A student was asked to simplify the expression $(3x^2 - 4x + 5) - [2x^2 - (5x - 3)(x + 2)]$. The student incorrectly dropped the outer brackets and distributed signs improperly, arriving at an erroneous result. Carefully evaluate the correct simplified polynomial in standard form.

A student was asked to simplify the expression $(3x^2 - 4x + 5) - [2x^2 - (5x - 3)(x + 2)]$ . The student incorrectly dropped the outer brackets and distributed signs improperly, arriving at an erroneous result. Carefully evaluate the correct simplified polynomial in standard form.

Final answer

$6x^2 + 3x - 1$

Steps

Expand the product inside the brackets

To simplify the expression $(3x^2 - 4x + 5) - [2x^2 - (5x - 3)(x + 2)]$ , follow the order of operations by working from the innermost grouping symbols outward.

First, expand $(5x - 3)(x + 2)$ using the distributive property (FOIL): $(5x - 3)(x + 2) = 5x(x) + 5x(2) - 3(x) - 3(2)$ $= 5x^2 + 10x - 3x - 6$ $= 5x^2 + 7x - 6$

Simplify inside the square brackets

Substitute the expanded product back inside the square brackets: $[2x^2 - (5x^2 + 7x - 6)]$

Distribute the negative sign across every term inside the parentheses: $2x^2 - 5x^2 - 7x + 6$

Combine like terms inside the brackets: $(2x^2 - 5x^2) - 7x + 6 = -3x^2 - 7x + 6$

Common Pitfall: When subtracting an expression in parentheses, the negative sign applies to every term inside, changing $+7x$ to $-7x$ and $-6$ to $+6$ .

Distribute the outer subtraction

Now, rewrite the full expression using the simplified bracket: $(3x^2 - 4x + 5) - [-3x^2 - 7x + 6]$

Distribute the outer negative sign to each term of the bracketed expression: $3x^2 - 4x + 5 - (-3x^2) - (-7x) - 6$ $= 3x^2 - 4x + 5 + 3x^2 + 7x - 6$

Combine like terms into standard form

Group terms with the same degree:

$x^2$ -terms: $3x^2 + 3x^2 = 6x^2$
$x$ -terms: $-4x + 7x = 3x$
Constant terms: $5 - 6 = -1$

Putting them together gives the polynomial in standard form: $6x^2 + 3x - 1$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the negative sign to all terms when expanding -(5x - 3)(x + 2), applying it only to the first term.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$(5x - 3)(x + 2) = 5x^2 + 7x - 6$
Step 2
$2x^2 - (5x^2 + 7x - 6) = 2x^2 - 5x^2 + 7x - 6 \implies -3x^2 + 7x - 6$
Correct
$2x^2 - (5x^2 + 7x - 6) = 2x^2 - 5x^2 - 7x + 6 \implies -3x^2 - 7x + 6$
When subtracting a polynomial in parentheses, the negative sign must be distributed to every term inside: $-(5x^2 + 7x - 6) = -5x^2 - 7x + 6$ .
Step 3
$(3x^2 - 4x + 5) - (-3x^2 + 7x - 6) \implies 3x^2 - 4x + 5 + 3x^2 - 7x + 6$
Correct
$(3x^2 - 4x + 5) - (-3x^2 - 7x + 6) \implies 3x^2 - 4x + 5 + 3x^2 + 7x - 6$
Your sign distribution here was performed logically based on your previous expression, but it carries forward the incorrect inner polynomial.
Step 4
$6x^2 - 11x + 11$
Correct
$6x^2 + 3x - 1$
Because of the sign error in the inner bracket, the final combined polynomial is incorrect.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly multiplied the binomials $(5x - 3)(x + 2)$ into $5x^2 + 7x - 6$ .

You followed the proper order of operations by working from innermost groupings outward.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign distribution error across parentheses

You applied the negative sign only to $5x^2$ and left $+7x - 6$ unchanged instead of flipping every sign inside the parentheses.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Incomplete distribution of a leading negative sign across multi-term expressions in parentheses.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute to every term: $-(a + b - c) = -a - b + c$
Treat a leading minus sign as multiplying every individual term inside the parentheses by $-1$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute the negative sign to all three terms: $2x^2 - (5x^2 + 7x - 6) = 2x^2 - 5x^2 - 7x + 6 = -3x^2 - 7x + 6$ .
Substitute back into the full expression: $(3x^2 - 4x + 5) - (-3x^2 - 7x + 6)$ .
Distribute the outer negative sign: $3x^2 - 4x + 5 + 3x^2 + 7x - 6$ .
Combine like terms: $(3x^2 + 3x^2) + (-4x + 7x) + (5 - 6) = 6x^2 + 3x - 1$ .

Final answer

$6x^2 + 3x - 1$

NEXT STEPS

Final Takeaways from Your Teacher

Draw distribution arrows or write an explicit ' $-1\cdot$ ' in front of parentheses when subtracting polynomials.
Double-check that every term inside parentheses flips its sign upon removing the grouping symbols.

OverviewACEpath Teacher's Diagnosis

02 The student drops the outer brackets without distributing the negative sign across all terms inside the brackets.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(5x - 3)(x + 2) = 5x^2 + 7x - 6
Step 2
2x^2 - (5x^2 + 7x - 6) = -3x^2 - 7x + 6
Step 3
(3x^2 - 4x + 5) - 3x^2 - 7x + 6
Correct
(3x^2 - 4x + 5) - (-3x^2 - 7x + 6) = 3x^2 - 4x + 5 + 3x^2 + 7x - 6
When subtracting the bracketed quantity $[-3x^2 - 7x + 6]$ , you must keep the grouping symbols and distribute the leading negative sign across every term: $-(-3x^2) = +3x^2$ , $-(-7x) = +7x$ , and $-(+6) = -6$ . Omitting the brackets caused two terms to retain the wrong signs and turned $+3x^2$ into $-3x^2$ .
Step 4
-11x + 11
Correct
6x^2 + 3x - 1
You combined like terms correctly according to your line 3, but the final polynomial is incorrect because of the sign distribution error in the previous step.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded the binomial product $(5x - 3)(x + 2)$ into $5x^2 + 7x - 6$ .

You accurately distributed the subtraction across the inner parentheses to obtain $-3x^2 - 7x + 6$ inside the brackets.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Dropped grouping symbols without distributing the negative sign

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the outer subtraction as if it only attached to the first term of the bracketed expression, dropping the brackets rather than distributing the negative factor $-1$ across every term of $(-3x^2 - 7x + 6)$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Retain Outer Brackets: When an entire multi-term expression is being subtracted, keep it enclosed in parentheses until the negative sign is explicitly distributed.
Write $-( -3x^2 - 7x + 6)$ instead of removing the brackets immediately.
Distribute -1 to Every Term: $-(a + b + c) = -a - b - c$ ; each internal sign flips.
Multiply every term inside by $-1$ : $-(-3x^2) = +3x^2$ , $-(-7x) = +7x$ , and $-(+6) = -6$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Rewrite the expression with brackets intact: $(3x^2 - 4x + 5) - (-3x^2 - 7x + 6)$ .
Distribute the negative sign: $3x^2 - 4x + 5 + 3x^2 + 7x - 6$ .
Group and combine like terms: $(3x^2 + 3x^2) + (-4x + 7x) + (5 - 6) = 6x^2 + 3x - 1$ .

Final answer

6x^2 + 3x - 1

NEXT STEPS

Final Takeaways from Your Teacher

Practice rewriting subtractions of bracketed polynomials by inserting an explicit ' $-1 \cdot (...)$ ' factor.
Always perform sign flipping as its own clear step before attempting to combine like terms.

OverviewACEpath Teacher's Diagnosis

03 The student adds the middle terms incorrectly when expanding (5x - 3)(x + 2) by multiplying coefficients instead of combining like terms.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(5x - 3)(x + 2) = 5x^2 + 13x - 6
Correct
(5x - 3)(x + 2) = 5x^2 + 7x - 6
When multiplying the two binomials, the outer product is $5x \cdot 2 = 10x$ and the inner product is $-3 \cdot x = -3x$ . Combining these gives $10x - 3x = 7x$ , not $13x$ .
Step 2
2x^2 - (5x^2 + 13x - 6) = -3x^2 - 13x + 6
Correct
2x^2 - (5x^2 + 7x - 6) = -3x^2 - 7x + 6
You distributed the negative sign through the parentheses and combined $2x^2 - 5x^2$ perfectly, but this line inherits the incorrect $+13x$ from the previous step.
Step 3
(3x^2 - 4x + 5) - (-3x^2 - 13x + 6) = 3x^2 - 4x + 5 + 3x^2 + 13x - 6
Correct
(3x^2 - 4x + 5) - (-3x^2 - 7x + 6) = 3x^2 - 4x + 5 + 3x^2 + 7x - 6
You handled the outer bracket subtraction flawlessly by flipping the signs of all terms, carrying forward the previous expression.
Step 4
6x^2 + 9x - 1
Correct
6x^2 + 3x - 1
You combined like terms correctly according to your previous line ( $-4x + 13x = 9x$ ), but because the linear term should have been $+7x$ , the correct middle term is $-4x + 7x = 3x$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You followed a clear order of operations by resolving the innermost product first.

You correctly distributed the negative sign across parentheses in both subtraction steps.

You grouped and combined like terms accurately in the final polynomial.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Linear term sign combination error in FOIL

In the expansion of $(5x - 3)(x + 2)$ , you combined $10x$ and $-3x$ as if both terms were positive, writing $13x$ instead of $7x$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking the negative sign attached to the constant term $-3$ when summing the outer and inner products ( $10x + (-3x)$ ).

REASONING PATH

Your ACEpath teacher’s suggested approach

Write Out the O and I Products Explicitly: For $(ax + b)(cx + d)$ , the linear term is $(ad + bc)x$ . Keep each term's sign intact.
Write $+10x - 3x$ as a separate intermediate line before combining them into $+7x$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand $(5x - 3)(x + 2) = 5x^2 + 10x - 3x - 6 = 5x^2 + 7x - 6$ .
Simplify inside the brackets: $2x^2 - (5x^2 + 7x - 6) = -3x^2 - 7x + 6$ .
Distribute the outer negative sign: $(3x^2 - 4x + 5) - (-3x^2 - 7x + 6) = 3x^2 - 4x + 5 + 3x^2 + 7x - 6$ .
Combine like terms: $(3x^2 + 3x^2) + (-4x + 7x) + (5 - 6) = 6x^2 + 3x - 1$ .

Final answer

$6x^2 + 3x - 1$

NEXT STEPS

Final Takeaways from Your Teacher

Practice writing out all four FOIL terms individually before combining the linear terms.
Highlight or circle negative signs in front of constants before multiplying binomials.

Lock it in

Question

What does a negative sign immediately in front of parentheses do to each term inside, such as in $-(3x - 4)$ ?

Answer

It multiplies every term inside by $-1$ , flipping all signs: $-3x + 4$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the negative sign to all terms when expanding -(5x - 3)(x + 2), applying it only to the first term.

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 drops the outer brackets without distributing the negative sign across all terms inside the brackets.

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 adds the middle terms incorrectly when expanding (5x - 3)(x + 2) by multiplying coefficients instead of combining like terms.

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

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