An open rectangular box is constructed from a rectangular sheet of cardboard whose length is $(5x + 4)$ centimeters and width is $(3x + 2)$ centimeters by cutting out identical square corners of side length $x$ centimeters and folding up the flaps. Express the total outer surface area $S(x)$ of the open box as a simplified polynomial in standard form by setting up the geometric decomposition and combining all like terms.

Final answer

$S(x) = 11x^2 + 22x + 8\text{ cm}^2$

Steps

To find the total outer surface area $S(x)$ of the open box, we can use the geometric decomposition of the flattened cardboard sheet.

Method 1: Subtraction of Corner Cutouts (Simplest Approach)

Calculate the area of the original rectangular sheet: The original dimensions are $(5x + 4)\text{ cm}$ by $(3x + 2)\text{ cm}$ . $\text{Area}_{\text{sheet}} = (5x + 4)(3x + 2)$

Expanding the product using the distributive property: $\text{Area}_{\text{sheet}} = 5x(3x) + 5x(2) + 4(3x) + 4(2)$ $\text{Area}_{\text{sheet}} = 15x^2 + 10x + 12x + 8$

Combining the like linear terms ( $10x + 12x$ ): $\text{Area}_{\text{sheet}} = 15x^2 + 22x + 8$
Calculate the area of the removed corners: Four square corners, each with side length $x\text{ cm}$ , are removed: $\text{Area}_{\text{cutouts}} = 4 \cdot (x \cdot x) = 4x^2$
Subtract the cutout area from the total area: Because the open box is formed strictly by folding up the remaining flaps, its outer surface area equals the area of the cardboard remaining after cutting: $S(x) = \text{Area}_{\text{sheet}} - \text{Area}_{\text{cutouts}}$ $S(x) = (15x^2 + 22x + 8) - 4x^2$
Combine like terms to write in standard form: Group the quadratic terms: $S(x) = (15x^2 - 4x^2) + 22x + 8$ $S(x) = 11x^2 + 22x + 8\text{ cm}^2$

Alternative Verification: Sum of the Five Faces

We can also find $S(x)$ by summing the area of the rectangular base and the four vertical sides:

Base length: $(5x + 4) - 2x = 3x + 4$
Base width: $(3x + 2) - 2x = x + 2$
Box height: $x$

$\text{Base Area} = (3x + 4)(x + 2) = 3x^2 + 10x + 8$ $\text{Area of two long sides} = 2 \cdot x(3x + 4) = 6x^2 + 8x$ $\text{Area of two short sides} = 2 \cdot x(x + 2) = 2x^2 + 4x$

Adding the base and sides together: $S(x) = (3x^2 + 10x + 8) + (6x^2 + 8x) + (2x^2 + 4x) = 11x^2 + 22x + 8\text{ cm}^2$

Common Pitfall: When finding the dimensions of the folded base, remember that a corner of width $x$ is removed from both ends of each dimension, so you must subtract $2x$ , not just $x$ , from the original length and width.

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student subtracts only one corner square instead of all four cut corners when calculating the remaining surface area.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
S(x) = (5x + 4)(3x + 2) - x^2
Correct
S(x) = (5x + 4)(3x + 2) - 4x^2
A rectangular sheet has 4 corners. Removing identical square corners of side length $x$ removes four squares with a total area of $4 \cdot x^2 = 4x^2$ , not just a single $x^2$ .
Step 2
S(x) = 15x^2 + 10x + 12x + 8 - x^2
Correct
S(x) = 15x^2 + 10x + 12x + 8 - 4x^2
Your expansion of $(5x + 4)(3x + 2)$ into $15x^2 + 10x + 12x + 8$ is completely correct, but the $-x^2$ term carries over the missing factor of 4 from Step 1.
Step 3
S(x) = 14x^2 + 22x + 8
Correct
S(x) = 11x^2 + 22x + 8
You accurately combined like terms according to your previous line ( $15x^2 - x^2 = 14x^2$ ), but the resulting coefficient is incorrect due to the initial setup error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly represented the total area of the original cardboard sheet as $(5x + 4)(3x + 2)$ .

You expanded the binomial product $(5x + 4)(3x + 2)$ into $15x^2 + 22x + 8$ without any algebraic slips.

You correctly grouped and combined like terms to write your expression in standard form.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omitted Multiplicity in Geometric Decomposition

You wrote $-x^2$ instead of $-4x^2$ , accounting for only one cutout corner instead of all four identical corners.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Focusing on the area of an individual corner unit ( $x \cdot x = x^2$ ) caused you to overlook the total quantity of corners being removed from the sheet.

REASONING PATH

Your ACEpath teacher’s suggested approach

Count All Subtracted Regions: Total cutout area equals $(\text{number of cutouts}) \times (\text{area of one cutout})$ .
Always sketch or annotate the rectangle to verify that all 4 corner cutouts are represented in the area subtraction: $4 \times x^2 = 4x^2$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Account for all 4 corners: $\text{Area}_{\text{cutouts}} = 4 \cdot (x \cdot x) = 4x^2$ .
Set up the remaining surface area: $S(x) = (5x + 4)(3x + 2) - 4x^2$ .
Expand the product: $S(x) = 15x^2 + 10x + 12x + 8 - 4x^2$ .
Combine like quadratic and linear terms: $S(x) = (15x^2 - 4x^2) + (10x + 12x) + 8 = 11x^2 + 22x + 8$ .

Final answer

S(x) = 11x^2 + 22x + 8 $\text{ cm}$ ^2

NEXT STEPS

Final Takeaways from Your Teacher

Before expanding expressions, annotate word problems with the count of identical shapes being removed.
Practice double-checking geometric decomposition formulas against a quick 2D sketch.

OverviewACEpath Teacher's Diagnosis

02 The student calculates the volume of the box instead of its outer surface area by multiplying the dimensions of the folded box.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\text{Length} = 5x + 4 - 2x = 3x + 4 \implies \text{Width} = 3x + 2 - 2x = x + 2 \implies \text{Height} = x$
Step 2
S(x) = x(3x + 4)(x + 2)
Correct
S(x) = (5x + 4)(3x + 2) - 4x^2
Multiplying $\text{height} \times \text{length} \times \text{width}$ computes the volume of the box. Surface area represents the combined 2D area of the cardboard sheet minus the four cut corners, or the sum of the bottom base and four vertical faces.
Step 3
S(x) = x(3x^2 + 10x + 8) $\implies$ S(x) = 3x^3 + 10x^2 + 8x
Correct
S(x) = (15x^2 + 22x + 8) - 4x^2 = 11x^2 + 22x + 8
Your polynomial expansion followed correctly from your formula, but the result is cubic (volume) rather than quadratic (surface area).

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly accounted for subtracting a corner of length $x$ from both ends of each dimension.

You accurately found the expressions for the base length, base width, and box height.

You carried out the polynomial multiplication and distribution accurately.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Formula confusion: volume instead of surface area

You set up $V = l \cdot w \cdot h$ instead of calculating the surface area $S(x)$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You confused the geometric concept of volume (a 3D cubic measure formed by multiplying three dimensions) with surface area (a 2D quadratic measure representing the total area of the material used).

REASONING PATH

Your ACEpath teacher’s suggested approach

Check the units and power: Area must be quadratic (degree 2 in this context, measured in $\text{cm}^2$ ), while volume is cubic (degree 3, measured in $\text{cm}^3$ ).
Whenever asked for surface area $S(x)$ , verify that your leading degree is $x^2$ , not $x^3$ .
Use net decomposition for folded cutouts: The outer surface area of a folded open box equals the area of the flat starting sheet minus the area of the removed corners.
Set up $S(x) = \text{Area}_{\text{sheet}} - 4 \cdot \text{Area}_{\text{corner}} = (5x + 4)(3x + 2) - 4(x^2)$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Find the total area of the original rectangular sheet: $(5x + 4)(3x + 2) = 15x^2 + 10x + 12x + 8 = 15x^2 + 22x + 8$ .
Find the area of the four removed square corners: $4 \cdot (x \cdot x) = 4x^2$ .
Subtract the removed corner area from the sheet area: $S(x) = (15x^2 + 22x + 8) - 4x^2$ .
Combine like terms to express in standard form: $S(x) = 11x^2 + 22x + 8$ .

Final answer

S(x) = 11x^2 + 22x + 8 $\text{ cm}$ ^2

NEXT STEPS

Final Takeaways from Your Teacher

Double-check whether a geometry problem asks for capacity/space inside (volume) or material needed/boundary coverage (surface area).
Practice using the flat net shortcut: total surface area of an open box formed by cut corners is simply the sheet area minus the cut corners.
Verify your final polynomial degree against the question type: area produces degree 2, volume produces degree 3.

OverviewACEpath Teacher's Diagnosis

03 The student includes a top lid by calculating the surface area of a closed rectangular prism with the folded dimensions.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\text{Base length} = 3x + 4 \implies \text{Base width} = x + 2 \implies \text{Height} = x$
Step 2
$S(x) = 2(3x + 4)(x + 2) + 2x(3x + 4) + 2x(x + 2)$
Correct
$S(x) = (3x + 4)(x + 2) + 2x(3x + 4) + 2x(x + 2)$
The problem specifies an open rectangular box, which has 5 faces (1 base and 4 sides). By multiplying the base area $(3x + 4)(x + 2)$ by 2, you included a non-existent top face.
Step 3
$S(x) = 2(3x^2 + 10x + 8) + 6x^2 + 8x + 2x^2 + 4x \implies S(x) = 6x^2 + 20x + 16 + 8x^2 + 12x$
Correct
$S(x) = (3x^2 + 10x + 8) + (6x^2 + 8x) + (2x^2 + 4x)$
Your polynomial expansion and term collection were carried out correctly, but they propagate the extra base factor from Step 2.
Step 4
$S(x) = 14x^2 + 32x + 16$
Correct
$S(x) = 11x^2 + 22x + 8$
The final polynomial is incorrect solely because of the added top face in Step 2.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly subtracted $2x$ from both dimensions to determine the dimensions of the base.

You correctly identified the height of the box as $x$ .

You accurately expanded the binomial products and combined like terms in your algebraic steps.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Extra face added to open box

You used the standard formula for a closed prism, doubling the base area even though the problem specifies an open box.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the surface area formula of a closed rectangular prism (6 faces: 2 bases and 4 lateral faces) with an open box (5 faces: 1 base and 4 lateral faces), or failing to use the simpler cardboard-subtraction method: $\text{Total Sheet Area} - 4(\text{Corner Cutouts})$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Check the number of faces: An open box has no lid, so it consists of exactly 5 faces (1 bottom base and 4 vertical sides).
Count the faces before writing the formula: write $\text{Base} + 2\text{Side}_1 + 2\text{Side}_2$ , never $2\times \text{Base}$ unless a lid is explicitly mentioned.
Use cardboard subtraction as a simpler check: Folding cardboard does not change the material's area.
Calculate $\text{Total Sheet Area} - \text{Cutouts Area} = (5x+4)(3x+2) - 4x^2$ to immediately find the outer surface area without splitting into individual faces.

CORRECTED PATH

ACEpath Teacher's Demonstration

Count 1 base and 4 vertical flaps: $S(x) = (3x + 4)(x + 2) + 2x(3x + 4) + 2x(x + 2)$ .
Expand each term: $S(x) = (3x^2 + 10x + 8) + (6x^2 + 8x) + (2x^2 + 4x)$ .
Group and combine like terms: $S(x) = (3x^2 + 6x^2 + 2x^2) + (10x + 8x + 4x) + 8$ .

Final answer

$S(x) = 11x^2 + 22x + 8$

NEXT STEPS

Final Takeaways from Your Teacher

Review the distinction between open containers (5 faces) and closed containers (6 faces).
Practice using the sheet-minus-cutout method for net-folding geometric problems to save time and reduce algebraic errors.

Lock it in

Question

When creating an open box by cutting square corners of side length $x$ from a rectangular sheet, what is the total area removed?

Answer

$4x^2$ . A rectangle has 4 corners, so four identical squares of area $x^2$ are removed.

Resources

Answer and steps

Final answer

Steps

Method 1: Subtraction of Corner Cutouts (Simplest Approach)

Alternative Verification: Sum of the Five Faces

Typical errors

01 The student subtracts only one corner square instead of all four cut corners when calculating the remaining surface area.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student calculates the volume of the box instead of its outer surface area by multiplying the dimensions of the folded box.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student includes a top lid by calculating the surface area of a closed rectangular prism with the folded dimensions.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

Similar question