Simplify the algebraic expression by expanding and combining all like terms: $4(3x^2 - 5x + 2) - 3(2x^2 - 7x - 6) + 2(x^2

Simplify the algebraic expression by expanding and combining all like terms: $4(3x^2 - 5x + 2) - 3(2x^2 - 7x - 6) + 2(x^2 - 4)$ .

Final answer

$8x^2 + x + 18$

Steps

Expand each term using the distributive property

To simplify the expression, we use the distributive property to remove the parentheses, and then combine like terms.

Multiply the factor outside each set of parentheses by every term inside:

Expand $4(3x^2 - 5x + 2)$ : $4 \cdot 3x^2 + 4 \cdot (-5x) + 4 \cdot 2 = 12x^2 - 20x + 8$
Expand $-3(2x^2 - 7x - 6)$ : Be careful to distribute the negative sign with the $3$ : $-3 \cdot 2x^2 - 3 \cdot (-7x) - 3 \cdot (-6) = -6x^2 + 21x + 18$

Common Pitfall: A frequent mistake is forgetting that multiplying two negative numbers yields a positive number. Here, $(-3) \cdot (-7x) = +21x$ and $(-3) \cdot (-6) = +18$ .
Expand $2(x^2 - 4)$ : $2 \cdot x^2 + 2 \cdot (-4) = 2x^2 - 8$

Write out the expanded expression

Combine the expanded parts into a single line: $12x^2 - 20x + 8 - 6x^2 + 21x + 18 + 2x^2 - 8$

Group and combine like terms

Group terms with the same degree (powers of $x$ ):

$x^2$ -terms: $(12 - 6 + 2)x^2 = 8x^2$
$x$ -terms: $(-20 + 21)x = 1x = x$
Constant terms: $8 + 18 - 8 = 18$

Final simplified expression

Putting all the combined terms together gives: $8x^2 + x + 18$

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, only applying it to the first term.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
12x^2 - 20x + 8 - 6x^2 - 21x - 18 + 2x^2 - 8
Correct
12x^2 - 20x + 8 - 6x^2 + 21x + 18 + 2x^2 - 8
When distributing $-3$ across $(2x^2 - 7x - 6)$ , multiplying a negative coefficient by negative terms produces positive terms: $(-3) \cdot (-7x) = +21x$ and $(-3) \cdot (-6) = +18$ , rather than $-21x$ and $-18$ .
Step 2
(12 - 6 + 2)x^2 + (-20 - 21)x + (8 - 18 - 8)
Correct
(12 - 6 + 2)x^2 + (-20 + 21)x + (8 + 18 - 8)
Your grouping technique was algebraically sound, but the coefficients carried over the incorrect signs from the previous expansion step.
Step 3
8x^2 - 41x - 18
Correct
8x^2 + x + 18
The final evaluation is mathematically consistent with your grouped terms, but incorrect due to the initial sign error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly distributed the positive multipliers $4$ and $2$ across their respective parentheses.

You correctly organized the expression by grouping terms with matching powers of $x$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive sign error with negative multiplier

You wrote $-21x - 18$ instead of $+21x + 18$ when expanding $-3(2x^2 - 7x - 6)$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Forgetting that multiplying a negative factor by a subtracted (negative) term flips the sign to positive, leading to the subtraction signs being carried through unchanged.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute the Negative Sign: For any terms $a, b, c$ , $-a(b - c) = -ab + ac$ .
Treat the minus sign in front of a bracket as part of the multiplier and multiply each inside term individually by $(-a)$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the full expression correctly: $4(3x^2 - 5x + 2) - 3(2x^2 - 7x - 6) + 2(x^2 - 4) = 12x^2 - 20x + 8 - 6x^2 + 21x + 18 + 2x^2 - 8$
Group coefficients of like terms: $(12 - 6 + 2)x^2 + (-20 + 21)x + (8 + 18 - 8)$
Simplify each group: $8x^2 + 1x + 18 = 8x^2 + x + 18$

Final answer

$8x^2 + x + 18$

NEXT STEPS

Final Takeaways from Your Teacher

Circle the negative multiplier and draw arrows to every term inside the parentheses before multiplying.
Double-check sign products explicitly: $(-)(-) = (+)$ and $(-)(+) = (-)$ .

OverviewACEpath Teacher's Diagnosis

02 The student fails to distribute the coefficient to the constant term inside the parentheses.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
12x^2 - 20x + 2 - 6x^2 + 21x - 6 + 2x^2 - 4
Correct
12x^2 - 20x + 8 - 6x^2 + 21x + 18 + 2x^2 - 8
When expanding each group with the distributive property, the outer multiplier must multiply every term inside the parentheses, including the constant. You wrote $+2$ , $-6$ , and $-4$ instead of $4 \cdot 2 = +8$ , $-3 \cdot (-6) = +18$ , and $2 \cdot (-4) = -8$ .
Step 2
(12 - 6 + 2)x^2 + (-20 + 21)x + (2 - 6 - 4) $\implies$ 8x^2 + x - 8
Correct
(12 - 6 + 2)x^2 + (-20 + 21)x + (8 + 18 - 8) $\implies$ 8x^2 + x + 18
You correctly grouped and combined the like terms based on your expanded line, but the result inherited the incorrect constant values from Step 1.
Step 3
8x^2 + x - 8
Correct
8x^2 + x + 18
Your final answer is affected by the missing distribution to the constant terms in Step 1.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly multiplied the outer coefficients with the variable terms $x^2$ and $x$ .

You handled the negative signs accurately when multiplying $-3$ by $2x^2$ and $-7x$ .

You correctly grouped like terms by degree and computed their coefficients accurately.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete distribution to constant term

You copied the constant terms directly from inside the parentheses without multiplying them by the factor outside.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Incomplete distribution across all terms inside parentheses. When expanding an expression of the form $a(bx^2 + cx + d)$ , the multiplier $a$ must be multiplied by every term inside, giving $abx^2 + acx + ad$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Count the Terms: If there are three terms inside the parentheses, expanding must produce three multiplied products.
Draw distribution arcs from the outer number to every term inside, especially the last constant term, before writing out the result.
Watch the Sign on Constants: Remember to include the sign when multiplying: $(-a) \cdot (-b) = +ab$ .
Treat the sign in front of a constant as part of that number when distributing.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute each coefficient across all terms inside the parentheses: $4(3x^2 - 5x + 2) = 12x^2 - 20x + 8$ , $-3(2x^2 - 7x - 6) = -6x^2 + 21x + 18$ , and $2(x^2 - 4) = 2x^2 - 8$ .
Write the expanded expression: $12x^2 - 20x + 8 - 6x^2 + 21x + 18 + 2x^2 - 8$ .
Group like terms: $(12 - 6 + 2)x^2 + (-20 + 21)x + (8 + 18 - 8)$ .
Combine the coefficients: $8x^2 + x + 18$ .

Final answer

8x^2 + x + 18

NEXT STEPS

Final Takeaways from Your Teacher

Practice drawing arrows from the outer coefficient to every term inside parentheses to ensure no term is missed.
Pay special attention to the constant term at the end of a polynomial inside parentheses.

OverviewACEpath Teacher's Diagnosis

03 The student incorrectly adds the exponents when combining like terms with variables.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
12x^2 - 20x + 8 - 6x^2 + 21x + 18 + 2x^2 - 8
Step 2
8x^4 + x^2 + 18
Correct
8x^2 + x + 18
When adding or subtracting like terms, only the coefficients are combined, while the variable and its exponent remain unchanged. Here, $(12 - 6 + 2)x^2 = 8x^2$ (not $8x^4$ ), and $(-20 + 21)x = 1x = x$ (not $x^2$ ).

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the distributive property to all three polynomial groups.

You handled the negative distribution $(-3) \cdot (-7x) = +21x$ and $(-3) \cdot (-6) = +18$ without any sign errors.

You combined the numerical coefficients and the constant terms accurately.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Adding exponents when combining like terms

This is a conceptual confusion between adding like terms and multiplying powers of the same base.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You applied the exponent product rule ( $x^a \cdot x^b = x^{a+b}$ ) to addition ( $ax^n + bx^n = (a+b)x^n$ ). In addition, like terms represent counts of identical units, so the unit ( $x^2$ or $x$ ) never changes its exponent.

REASONING PATH

Your ACEpath teacher’s suggested approach

Identify the operation: Are you adding terms or multiplying terms?
Use $ax^n + bx^n = (a+b)x^n$ for addition, and $(ax^m)(bx^n) = (ab)x^{m+n}$ for multiplication.
Preserve the unit: Like terms share the exact same variable part and exponent.
Treat $x^2$ like an item name: $12\text{ boxes} - 6\text{ boxes} + 2\text{ boxes} = 8\text{ boxes}$ , meaning $8x^2$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Group like terms: $(12x^2 - 6x^2 + 2x^2) + (-20x + 21x) + (8 + 18 - 8)$
Combine coefficients of $x^2$ : $(12 - 6 + 2)x^2 = 8x^2$
Combine coefficients of $x$ : $(-20 + 21)x = x$
Combine constant terms: $8 + 18 - 8 = 18$
Write the final sum: $8x^2 + x + 18$

Final answer

$8x^2 + x + 18$

NEXT STEPS

Final Takeaways from Your Teacher

Review the distinction between $x^2 + x^2 = 2x^2$ and $x^2 \cdot x^2 = x^4$ .
Complete the flashcards below to lock in the rule for combining like terms.
Solve the practice problems by checking off the power on each term after combining.

Lock it in

Question

What is the result of expanding $-a(b - c)$ ?

Answer

$-ab + ac$ . Distributing a negative number across a subtracted term turns that term positive because $(-a) \cdot (-c) = +ac$ .

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, only applying it 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 fails to distribute the coefficient to the constant term inside the parentheses.

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 incorrectly adds the exponents when combining like terms with variables.

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