Carefully expand and simplify the following expression, paying strict attention to signs and nested grouping: $-2(4x - 3y)^2 - [-(4x + 3y)(4x - 3y)]$. Express your final polynomial in standard form.

Carefully expand and simplify the following expression, paying strict attention to signs and nested grouping: $-2(4x - 3y)^2 - [-(4x + 3y)(4x - 3y)]$ . Express your final polynomial in standard form.

Final answer

-16x^2 + 48xy - 27y^2

Steps

Expand the first term using the square of a binomial pattern

To expand and simplify the expression, we break it down using special product formulas and carefully track the signs in the nested groupings.

Recall the identity for the square of a binomial: $(a - b)^2 = a^2 - 2ab + b^2$

Apply this pattern to $(4x - 3y)^2$ with $a = 4x$ and $b = 3y$ : $(4x - 3y)^2 = (4x)^2 - 2(4x)(3y) + (3y)^2 = 16x^2 - 24xy + 9y^2$

Now multiply this result by the outer coefficient, $-2$ : $-2(16x^2 - 24xy + 9y^2) = -32x^2 + 48xy - 18y^2$

Common Pitfall: When distributing a negative number such as $-2$ , ensure every term inside the parentheses changes sign. A frequent error is forgetting to change the sign of the middle term: $-2 \cdot (-24xy) = +48xy$ .

Simplify the second term using the difference of squares pattern

Look at the nested grouping: $-[-(4x + 3y)(4x - 3y)]$

First, resolve the outer negative signs. Since the negative of a negative is positive, $-[-A] = +A$ : $-[-(4x + 3y)(4x - 3y)] = +(4x + 3y)(4x - 3y)$

Next, apply the difference of squares identity, $(a + b)(a - b) = a^2 - b^2$ : $(4x + 3y)(4x - 3y) = (4x)^2 - (3y)^2 = 16x^2 - 9y^2$

Combine like terms

Substitute the simplified parts back into the complete expression: $(-32x^2 + 48xy - 18y^2) + (16x^2 - 9y^2)$

Group and combine like terms:

For $x^2$ : $-32x^2 + 16x^2 = -16x^2$
For $xy$ : $48xy$
For $y^2$ : $-18y^2 - 9y^2 = -27y^2$

Writing the polynomial in standard form gives: $-16x^2 + 48xy - 27y^2$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student squares each term inside the parentheses individually and omits the middle cross-term -24xy when expanding (4x - 3y)².

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
-2(16x^2 + 9y^2) - [-(16x^2 - 9y^2)]
Correct
-2(16x^2 - 24xy + 9y^2) - [-(16x^2 - 9y^2)]
When expanding $(4x - 3y)^2$ , you squared each term individually as $(4x)^2 + (-3y)^2 = 16x^2 + 9y^2$ , missing the cross-term $-2(4x)(3y) = -24xy$ from the identity $(a - b)^2 = a^2 - 2ab + b^2$ .
Step 2
-32x^2 - 18y^2 + 16x^2 - 9y^2
Correct
-32x^2 + 48xy - 18y^2 + 16x^2 - 9y^2
Your distribution of $-2$ and resolution of the nested signs $-[-(\dots)] = +(\dots)$ were executed correctly based on your previous expression, but the term $+48xy$ is missing due to the earlier error.
Step 3
-16x^2 - 27y^2
Correct
-16x^2 + 48xy - 27y^2
You accurately combined like terms $-32x^2 + 16x^2 = -16x^2$ and $-18y^2 - 9y^2 = -27y^2$ , but the final polynomial lacks the middle term $48xy$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the difference of squares identity $(4x + 3y)(4x - 3y) = 16x^2 - 9y^2$ .

You handled the double negation $-[-(\dots)]$ accurately to obtain a positive term.

You correctly distributed $-2$ across your terms and collected like terms without arithmetic mistakes.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omitted cross-term in binomial square

none

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the square of a binomial $(a - b)^2$ as if exponents distribute over subtraction, writing $a^2 + b^2$ instead of including the cross-term $-2ab$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Square of a Binomial Identity: $(a \pm b)^2 = a^2 \pm 2ab + b^2$
Always write down the three-term expansion before multiplying by outer coefficients.
Cross-term Verification: The middle term of $(4x - 3y)^2$ is $2 \cdot (4x) \cdot (-3y) = -24xy$ .
Multiply the two terms together and double them to guarantee the middle term is not omitted.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand both special products properly: $-2(16x^2 - 24xy + 9y^2) - [-(16x^2 - 9y^2)]$
Distribute $-2$ and simplify the double negative: $-32x^2 + 48xy - 18y^2 + (16x^2 - 9y^2)$
Combine like terms: $(-32x^2 + 16x^2) + 48xy + (-18y^2 - 9y^2) = -16x^2 + 48xy - 27y^2$

Final answer

-16x^2 + 48xy - 27y^2

NEXT STEPS

Final Takeaways from Your Teacher

Review the special product formulas, contrasting $(a - b)^2 = a^2 - 2ab + b^2$ with $(a - b)(a + b) = a^2 - b^2$ .
When expanding $(a - b)^2$ , draw arrows or write out the intermediate cross-term $2(a)(-b)$ explicitly before distributing external constants.

OverviewACEpath Teacher's Diagnosis

02 The student mishandles the triple negative in the second term, simplifying -[-(4x + 3y)(4x - 3y)] to -(16x² - 9y²) instead of +(16x² - 9y²).

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
-2(16x^2 - 24xy + 9y^2) - (16x^2 - 9y^2)
Correct
-2(16x^2 - 24xy + 9y^2) + (16x^2 - 9y^2)
You correctly applied the square of a binomial to get $(4x - 3y)^2 = 16x^2 - 24xy + 9y^2$ and the difference of squares to get $(4x + 3y)(4x - 3y) = 16x^2 - 9y^2$ . However, simplifying the nested negative signs $-[-(\dots)]$ must yield $+(\dots)$ , not $-(\dots)$ .
Step 2
-32x^2 + 48xy - 18y^2 - 16x^2 + 9y^2
Correct
-32x^2 + 48xy - 18y^2 + 16x^2 - 9y^2
You correctly distributed $-2$ into the first trinomial and consistently distributed the negative sign across your written expression $-(16x^2 - 9y^2)$ , but this step inherits the sign error from Step 1.
Step 3
-48x^2 + 48xy - 9y^2
Correct
-16x^2 + 48xy - 27y^2
You correctly combined like terms based on your previous line ( $-32x^2 - 16x^2 = -48x^2$ and $-18y^2 + 9y^2 = -9y^2$ ), but the resulting polynomial is incorrect due to the propagated sign error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded the perfect square binomial $(4x - 3y)^2$ to $16x^2 - 24xy + 9y^2$ .

You correctly applied the difference of squares identity $(4x + 3y)(4x - 3y) = 16x^2 - 9y^2$ .

You accurately distributed the negative coefficient $-2$ across all three terms of the first expansion.

You combined like terms accurately according to your working expressions.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign Error in Nested Grouping

Simplifying $-[-(\dots)]$ to $-(\dots)$ instead of $+(\dots)$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Losing track of paired negative signs when resolving nested grouping symbols (brackets and parentheses). Two consecutive negative signs outside an expression cancel out to a positive sign: $-[-P] = +P$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Double Negative Cancellation: Two negative signs multiplying or acting consecutively on an expression cancel to positive: $-[-P] = +P$ .
Before expanding inner products, simplify the signs of outer nested brackets first.

CORRECTED PATH

ACEpath Teacher's Demonstration

Replace $-[-(4x + 3y)(4x - 3y)]$ with $+ (4x + 3y)(4x - 3y) = +(16x^2 - 9y^2)$ .
Combine with the distributed first term: $(-32x^2 + 48xy - 18y^2) + (16x^2 - 9y^2)$ .
Group like terms: $(-32 + 16)x^2 + 48xy + (-18 - 9)y^2$ .

Final answer

-16x^2 + 48xy - 27y^2

NEXT STEPS

Final Takeaways from Your Teacher

Always simplify outer nested negative signs before removing brackets: replace $-[-(\dots)]$ with $+(\dots)$ immediately.
Circle or highlight outer sign pairs before carrying out binomial multiplications to avoid dropping a negative.

OverviewACEpath Teacher's Diagnosis

03 The student distributes -2 inside the parentheses before squaring instead of following the order of operations, squaring (-8x + 6y).

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(-8x + 6y)^2 + (16x^2 - 9y^2)
Correct
-2(16x^2 - 24xy + 9y^2) + (16x^2 - 9y^2)
According to the order of operations, exponents must be evaluated before multiplication. You cannot distribute $-2$ into the binomial $(4x - 3y)$ while it is raised to the second power. Instead, expand $(4x - 3y)^2$ first, then multiply each term by $-2$ .
Step 2
64x^2 - 96xy + 36y^2 + 16x^2 - 9y^2
Correct
-32x^2 + 48xy - 18y^2 + 16x^2 - 9y^2
You expanded your binomial $(-8x + 6y)^2$ correctly according to algebraic rules, but the resulting values carry forward the error from distributing prematurely in Step 1.
Step 3
80x^2 - 96xy + 27y^2
Correct
-16x^2 + 48xy - 27y^2
You grouped and combined like terms accurately ( $64x^2 + 16x^2 = 80x^2$ and $36y^2 - 9y^2 = 27y^2$ ), but the final polynomial is incorrect due to the initial distribution error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly resolved the double negative $-[-(\dots)]$ into addition.

You accurately applied the difference of squares identity to expand $(4x + 3y)(4x - 3y) = 16x^2 - 9y^2$ .

You correctly combined like terms in your final calculation.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Order of Operations: Exponents Before Multiplication

Distributing an outside coefficient into a base before applying the power.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the algebraic structure $k(A)^2$ with $(kA)^2$ . Multiplying the base by $k$ before squaring inadvertently scales the entire expression by $k^2$ instead of $k$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Exponent Hierarchy Check: In $c(ax + b)^n$ , exponentiation has precedence over multiplication by $c$ .
Always expand $(ax + b)^n$ completely in parentheses before distributing the outer coefficient $c$ .
Factor Scaling Sanity Check: $(c \cdot A)^2 = c^2 A^2 \neq c A^2$ .
Notice that $(-2 \cdot (4x - 3y))^2 = 4(4x - 3y)^2$ , which quadruples the expression instead of multiplying it by $-2$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the squared binomial first: $(4x - 3y)^2 = 16x^2 - 24xy + 9y^2$ .
Distribute the outer factor $-2$ : $-2(16x^2 - 24xy + 9y^2) = -32x^2 + 48xy - 18y^2$ .
Add the simplified second term: $(-32x^2 + 48xy - 18y^2) + (16x^2 - 9y^2) = -16x^2 + 48xy - 27y^2$ .

Final answer

-16x^2 + 48xy - 27y^2

NEXT STEPS

Final Takeaways from Your Teacher

Practice identifying outer coefficients versus exponents, ensuring you evaluate powers before distributing scalars.
When dealing with expressions of the form $k(a - b)^2$ , keep large brackets around the expanded polynomial until you perform the distribution step.

Lock it in

Question

What is the complete formula for the square of a binomial $(a - b)^2$ ?

Answer

$a^2 - 2ab + b^2$ . A squared binomial always produces a trinomial; remember the middle cross-term $-2ab$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student squares each term inside the parentheses individually and omits the middle cross-term -24xy when expanding (4x - 3y)².

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 mishandles the triple negative in the second term, simplifying -[-(4x + 3y)(4x - 3y)] to -(16x² - 9y²) instead of +(16x² - 9y²).

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 inside the parentheses before squaring instead of following the order of operations, squaring (-8x + 6y).

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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