Use the distributive property to expand and simplify the algebraic expression: $4x(3x^2 - 5x + 7) - 2(6x^3 - 8x^2 + 5x

Use the distributive property to expand and simplify the algebraic expression: $4x(3x^2 - 5x + 7) - 2(6x^3 - 8x^2 + 5x - 9)$ .

Final answer

$-4x^2 + 18x + 18$

Steps

Expand the first group

To expand and simplify the expression, apply the distributive property to each grouped set of terms and then combine like terms.

Multiply $4x$ by each term inside the first set of parentheses: $4x(3x^2 - 5x + 7) = (4x \cdot 3x^2) + (4x \cdot (-5x)) + (4x \cdot 7)$ $= 12x^3 - 20x^2 + 28x$

Expand the second group

Distribute $-2$ to each term inside the second set of parentheses. Remember to include the negative sign with the multiplier, which flips the sign of every term inside: $-2(6x^3 - 8x^2 + 5x - 9) = (-2 \cdot 6x^3) + (-2 \cdot (-8x^2)) + (-2 \cdot 5x) + (-2 \cdot (-9))$ $= -12x^3 + 16x^2 - 10x + 18$

Common Pitfall: Forgetting to distribute the negative sign to all terms is a frequent mistake. Specifically, ensure that $-2 \cdot (-8x^2) = +16x^2$ and $-2 \cdot (-9) = +18$ .

Combine all terms

Write the expanded expression in full: $(12x^3 - 20x^2 + 28x) + (-12x^3 + 16x^2 - 10x + 18)$

Group like terms according to the powers of $x$ :

Cubic terms ( $x^3$ ): $12x^3 - 12x^3 = 0$
Quadratic terms ( $x^2$ ): $-20x^2 + 16x^2 = -4x^2$
Linear terms ( $x$ ): $28x - 10x = 18x$
Constant terms: $+18$

Sum the grouped terms: $-4x^2 + 18x + 18$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the negative sign to all terms inside the second set of parentheses, resulting in incorrect signs for the remaining terms.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
= 12x^3 - 20x^2 + 28x - 12x^3 - 16x^2 + 5x - 9
Correct
12x^3 - 20x^2 + 28x - 12x^3 + 16x^2 - 10x + 18
Expanding $4x(3x^2 - 5x + 7)$ gives $12x^3 - 20x^2 + 28x$ , which you did perfectly. However, distributing $-2$ across each term of $(6x^3 - 8x^2 + 5x - 9)$ requires multiplying every term by $-2$ : $(-2)(6x^3) = -12x^3$ , $(-2)(-8x^2) = +16x^2$ , $(-2)(5x) = -10x$ , and $(-2)(-9) = +18$ .
Step 2
= (12 - 12)x^3 + (-20 - 16)x^2 + (28 + 5)x - 9
Correct
(12 - 12)x^3 + (-20 + 16)x^2 + (28 - 10)x + 18
Your grouping method by variable degree was completely logical, but it inherited the incorrect coefficients from the expansion step.
Step 3
= -36x^2 + 33x - 9
Correct
-4x^2 + 18x + 18
Your final calculation correctly evaluated your grouped line, but the result differs from $-4x^2 + 18x + 18$ due to the earlier distribution mistake.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded the first polynomial using the distributive property.

You grouped like terms systematically by descending powers of $x$ .

You carried out downstream arithmetic accurately according to your written grouping.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete distribution of a negative factor

You only multiplied the first two terms by 2, and you missed the sign flip on the second, third, and fourth terms when distributing $-2$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating the leading negative sign as an isolated minus symbol or stopping the distribution mid-parenthesis prevents every interior term from being multiplied by the full factor $-2$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Box the entire multiplier: The multiplier in front of $(6x^3 - 8x^2 + 5x - 9)$ includes the minus sign: it is the single entity $(-2)$ .
Mentally or visibly circle $-2$ and draw distribution arrows from it to all four interior terms.
Check each product individually: Every single interior term must have its coefficient multiplied by $-2$ , and like signs produce a positive result.
Write out the mini-products: $(-2)(-8x^2) = +16x^2$ , $(-2)(+5x) = -10x$ , and $(-2)(-9) = +18$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Rewrite the full expansion with correct signs: $12x^3 - 20x^2 + 28x - 12x^3 + 16x^2 - 10x + 18$
Combine like terms: $(12 - 12)x^3 + (-20 + 16)x^2 + (28 - 10)x + 18$
Simplify each coefficient: $0x^3 - 4x^2 + 18x + 18$

Final answer

-4x^2 + 18x + 18

NEXT STEPS

Final Takeaways from Your Teacher

Practice distributing negative numbers across polynomials with 3 or more terms.
Always count the terms inside the parentheses to verify that your expanded line has the exact same number of resulting products.

OverviewACEpath Teacher's Diagnosis

02 The student multiplies the variable by the coefficient but fails to add the exponents of x, treating 4x · 3x² as 12x² instead of 12x³.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
12x^2 - 20x + 28x - 12x^3 + 16x^2 - 10x + 18
Correct
12x^3 - 20x^2 + 28x - 12x^3 + 16x^2 - 10x + 18
When expanding $4x(3x^2 - 5x + 7)$ , you multiplied the numerical coefficients but omitted the variable factor from the exponent addition: $4x \cdot 3x^2 = 12x^3$ (not $12x^2$ ) and $4x \cdot (-5x) = -20x^2$ (not $-20x$ ). Your expansion of the second group, $-2(6x^3 - 8x^2 + 5x - 9) = -12x^3 + 16x^2 - 10x + 18$ , was completely correct.
Step 2
-12x^3 + (12 + 16)x^2 + (-20 + 28 - 10)x + 18 $\implies$ -12x^3 + 28x^2 - 2x + 18
Correct
(12 - 12)x^3 + (-20 + 16)x^2 + (28 - 10)x + 18 $\implies$ -4x^2 + 18x + 18
You correctly grouped and combined the like terms from your previous line, but the resulting polynomial was affected by the exponent error in Step 1.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly distributed the negative factor $-2$ to every term in the second set of parentheses, including tracking all sign changes.

You correctly identified and combined like terms according to their powers in your second line.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Exponent addition omitted during monomial distribution

Multiplying $4x$ by a polynomial while leaving the power of $x$ unchanged on the first two terms ( $4x \cdot 3x^2 \to 12x^2$ and $4x \cdot 5x \to 20x$ ).

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating the outside variable factor $x$ as if it were only a scalar multiplier, forgetting to apply the product rule of exponents $x^1 \cdot x^n = x^{1+n}$ to each internal term.

REASONING PATH

Your ACEpath teacher’s suggested approach

Apply the Product Rule for Exponents: When multiplying terms with the same base, add their powers: $x^a \cdot x^b = x^{a+b}$ . Specifically, $x^1 \cdot x^2 = x^3$ and $x^1 \cdot x^1 = x^2$ .
Write $x$ as $x^1$ explicitly before distributing to remind yourself to increase each exponent by $1$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Group like terms by degree: $(12x^3 - 12x^3) + (-20x^2 + 16x^2) + (28x - 10x) + 18$ .
Simplify each coefficient: $0x^3 - 4x^2 + 18x + 18$ .

Final answer

-4x^2 + 18x + 18

NEXT STEPS

Final Takeaways from Your Teacher

Always write $x^1$ on single-variable factors outside parentheses before distributing.
After expanding, do a quick degree check: each term inside the parentheses should increase in degree by the degree of the monomial outside.

OverviewACEpath Teacher's Diagnosis

03 The student distributes 2 instead of -2 across the second polynomial and subtracts only the first term, leaving the rest of the expanded terms added as positive products.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
12x^3 - 20x^2 + 28x - 12x^3 + 16x^2 + 10x - 18
Correct
12x^3 - 20x^2 + 28x - 12x^3 + 16x^2 - 10x + 18
You expanded $4x(3x^2 - 5x + 7)$ correctly to $12x^3 - 20x^2 + 28x$ . However, when distributing $-2$ across $(6x^3 - 8x^2 + 5x - 9)$ , you did not carry the negative sign to the last two terms: $(-2) \cdot 5x = -10x$ and $(-2) \cdot (-9) = +18$ , resulting in $+10x - 18$ instead.
Step 2
(12 - 12)x^3 + (-20 + 16)x^2 + (28 + 10)x - 18
Correct
(12 - 12)x^3 + (-20 + 16)x^2 + (28 - 10)x + 18
You correctly collected like terms by power of $x$ based on your expanded expression, but the linear and constant coefficients were affected by the sign error in the previous step.
Step 3
-4x^2 + 38x - 18
Correct
-4x^2 + 18x + 18
Your arithmetic evaluation of each bracketed coefficient was accurate, but the final expression carries over the sign errors from Step 1.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the distributive property to expand $4x(3x^2 - 5x + 7)$ into $12x^3 - 20x^2 + 28x$ .

You grouped like terms methodically using coefficient parentheses.

You evaluated the arithmetic in each grouped term accurately according to your written coefficients.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributing Negative Sign

You distributed $-2$ to the first two terms as $-12x^3$ and $+16x^2$ , but switched to multiplying by $+2$ for the remaining terms $+5x$ and $-9$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Incomplete distribution of a negative coefficient across longer polynomials, where sign inversion is forgotten halfway through the expression.

REASONING PATH

Your ACEpath teacher’s suggested approach

Attach the sign to the multiplier: When expanding $-(a \dots)$ or $-c(a + b)$ , treat the multiplier as $(-c)$ for every single term inside.
Write out $(-2) \cdot (5x)$ and $(-2) \cdot (-9)$ explicitly or draw distribution arcs to all four terms before calculating.
Count term signs: Multiplying by a negative factor must flip the sign of every individual term inside the parentheses.
Check each term's sign after distributing: positive terms must become negative, and negative terms must become positive.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the second group: $(-2)(6x^3) + (-2)(-8x^2) + (-2)(5x) + (-2)(-9) = -12x^3 + 16x^2 - 10x + 18$
Combine with the first group: $(12x^3 - 12x^3) + (-20x^2 + 16x^2) + (28x - 10x) + 18$
Simplify each coefficient: $0x^3 - 4x^2 + 18x + 18$

Final answer

-4x^2 + 18x + 18

NEXT STEPS

Final Takeaways from Your Teacher

Review the three flashcards below on distributing negative signs across multi-term polynomials.
Complete the three targeted practice problems to build consistency when handling negative multipliers.
Practice circling the negative sign together with the coefficient before distributing.

Lock it in

Question

What are the two components of the distributive property when expanding $-k(a - b + c)$ ?

Answer

Multiply every term inside by $k$ , and flip the sign of every term inside: $-k(a - b + c) = -ka + kb - kc$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the negative sign to all terms inside the second set of parentheses, resulting in incorrect signs for the remaining terms.

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 multiplies the variable by the coefficient but fails to add the exponents of x, treating 4x · 3x² as 12x² instead of 12x³.

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 distributes 2 instead of -2 across the second polynomial and subtracts only the first term, leaving the rest of the expanded terms added as positive products.

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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