Carefully expand and simplify the following algebraic expression, paying strict attention to negative signs and subtraction across nested grouping symbols: $-3x \left[ 2x - 4(x - 5) \right] - 2(x - 3)(5 - x)$. Express the final simplified polynomial in standard form $ax^2 + bx + c$.

Carefully expand and simplify the following algebraic expression, paying strict attention to negative signs and subtraction across nested grouping symbols: $-3x \left[ 2x - 4(x - 5) \right] - 2(x - 3)(5 - x)$ . Express the final simplified polynomial in standard form $ax^2 + bx + c$ .

Final answer

$8x^2 - 76x + 30$

Steps

Simplify the first part, $-3x \left[ 2x - 4(x - 5) \right]$

To simplify the expression, break it into two main parts, work from the inside out using the distributive property, and combine like terms.

Expand inside the brackets first:
Distribute the $-4$ across $(x - 5)$ : $-4(x - 5) = -4x + 20$

Substitute this back into the brackets and combine like terms: $2x - 4(x - 5) = 2x - 4x + 20 = -2x + 20$
Distribute the outer factor $-3x$ : $-3x(-2x + 20) = (-3x) \cdot (-2x) + (-3x) \cdot (20) = 6x^2 - 60x$

Common Pitfall: Watch out for the negative signs. Distributing $-4$ to $-5$ results in $+20$ , not $-20$ . Similarly, multiplying $-3x$ by $-2x$ results in $+6x^2$ .

Simplify the second part, $-2(x - 3)(5 - x)$

Expand the product of the two binomials $(x - 3)(5 - x)$ :
Use the distributive property (FOIL): $(x - 3)(5 - x) = x(5) + x(-x) - 3(5) - 3(-x)$ $= 5x - x^2 - 15 + 3x$ $= -x^2 + 8x - 15$
Multiply by the leading coefficient $-2$ : $-2(-x^2 + 8x - 15) = (-2) \cdot (-x^2) + (-2) \cdot (8x) + (-2) \cdot (-15)$ $= 2x^2 - 16x + 30$

Combine all terms

Now, add the simplified results from Step 1 and Step 2 together: $(6x^2 - 60x) + (2x^2 - 16x + 30)$

Group the like terms by degree:

$x^2$ -terms: $6x^2 + 2x^2 = 8x^2$
$x$ -terms: $-60x - 16x = -76x$
Constant terms: $+30$

Putting them together in standard form ( $ax^2 + bx + c$ ): $8x^2 - 76x + 30$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student fails to distribute the negative sign to the second term inside the parentheses, writing -4(x - 5) as -4x - 20 instead of -4x + 20.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
-3x[2x - 4x - 20] - 2(5x - x^2 - 15 + 3x)
Correct
-3x[2x - 4x + 20] - 2(5x - x^2 - 15 + 3x)
You expanded the product $(x - 3)(5 - x)$ accurately, but when distributing $-4$ into $(x - 5)$ , multiplying two negative numbers gives a positive result: $(-4) \cdot (-5) = +20$ .
Step 2
-3x[-2x - 20] - 2(-x^2 + 8x - 15)
Correct
-3x[-2x + 20] - 2(-x^2 + 8x - 15)
You correctly combined $2x - 4x$ to get $-2x$ and $5x + 3x$ to get $8x$ , but the expression carried forward the previous sign error from $-20$ .
Step 3
6x^2 + 60x + 2x^2 - 16x + 30
Correct
6x^2 - 60x + 2x^2 - 16x + 30
You applied the distributive property correctly to your previous line: $(-3x)(-2x) = 6x^2$ and $-2(-x^2 + 8x - 15) = 2x^2 - 16x + 30$ . Multiplying $-3x$ by $+20$ in the correct expression yields $-60x$ instead of $+60x$ .
Step 4
8x^2 + 44x + 30
Correct
8x^2 - 76x + 30
You combined like terms correctly according to your line ( $6x^2 + 2x^2 = 8x^2$ , $60x - 16x = 44x$ ), but the result is affected by the early sign error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded the binomial product $(x - 3)(5 - x)$ using FOIL.

You distributed the outer negative factors $-3x$ and $-2$ accurately across their respective parentheses.

You successfully grouped and combined like terms by degree into standard polynomial form.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive sign error with negative factors

You treated the subtraction sign in $(x - 5)$ as remaining negative after multiplying by the negative coefficient $-4$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking that distributing a negative multiplier changes the sign of every term inside the parentheses, specifically that $(-a)(x - b) = -ax + ab$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Attach the sign to the multiplier: Treat the minus sign in front of a coefficient as part of the number: $-c(A - B) = (-c)(A) + (-c)(-B)$ .
Circle or box the negative coefficient together with its sign before distributing across the terms inside.
Double-negative product check: Multiplying a negative constant by a subtracted term yields a positive term: $(-)(-) = (+)$ .
Whenever you distribute across a subtraction symbol, verify that the resulting sign flips to addition.

CORRECTED PATH

ACEpath Teacher's Demonstration

Correct the inner distribution: $-4(x - 5) = -4x + 20$ .
Simplify inside the brackets: $2x - 4x + 20 = -2x + 20$ .
Distribute $-3x$ into the brackets: $-3x(-2x + 20) = 6x^2 - 60x$ .
Add the second distributed part: $(6x^2 - 60x) + (2x^2 - 16x + 30)$ .
Combine like terms: $(6x^2 + 2x^2) + (-60x - 16x) + 30 = 8x^2 - 76x + 30$ .

Final answer

8x^2 - 76x + 30

NEXT STEPS

Final Takeaways from Your Teacher

Practice distributing negative numbers across expressions containing subtraction symbols.
Write out intermediate signs explicitly, such as $+ (-4)(-5)$ , before combining into a single operation.
Perform a quick sign audit on all constant terms before moving to the next algebraic line.

OverviewACEpath Teacher's Diagnosis

02 The student expands the product 2(x - 3)(5 - x) but subtracts only the first term rather than distributing the negative sign across all terms.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
=-3x[2x - 4x + 20] - 2(-x^2 + 8x - 15) $\implies$ -3x[-2x + 20] - (-2x^2 + 16x - 30)
Step 2
=6x^2 - 60x + 2x^2 + 16x - 30
Correct
6x^2 - 60x + 2x^2 - 16x + 30
When expanding $-(-2x^2 + 16x - 30)$ , you applied the negative sign to $-2x^2$ to get $+2x^2$ , but you did not change the signs of $+16x$ and $-30$ . Distributing the negative sign across all three terms yields $+2x^2 - 16x + 30$ .
Step 3
=8x^2 - 44x - 30
Correct
8x^2 - 76x + 30
You combined like terms consistently according to your previous line, but because of the sign error in the previous step, your linear and constant terms are incorrect.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the distributive property to inner terms like $-4(x - 5)$ and handled the negative multiplication to obtain $+20$ .

You expanded the binomial product $(x - 3)(5 - x)$ to $-x^2 + 8x - 15$ without errors.

You combined like terms correctly based on the coefficients written in your third step.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributing Negative Signs Across Parentheses

You distributed the leading negative sign only to the first term of $(-2x^2 + 16x - 30)$ , writing $+16x - 30$ instead of $-16x + 30$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating a minus sign preceding a group as applying only to the leading term rather than multiplying every term inside the grouping symbol by $-1$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Negating a Polynomial: $-(a + b - c) = -a - b + c$
Change the sign of every single term inside the parentheses when removing them.
Alternative Strategy: Distribute $-2$ Directly: $-2(a + b + c) = (-2)(a) + (-2)(b) + (-2)(c)$
Multiply each term directly by the signed coefficient $-2$ in one step to avoid managing a separate outer minus sign.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute $-3x$ to the first group: $(-3x)(-2x) + (-3x)(20) = 6x^2 - 60x$
Distribute the negative sign to all terms in the second group: $-(-2x^2) - (16x) - (-30) = 2x^2 - 16x + 30$
Combine the expressions: $(6x^2 - 60x) + (2x^2 - 16x + 30)$
Collect like terms: $(6x^2 + 2x^2) + (-60x - 16x) + 30 = 8x^2 - 76x + 30$

Final answer

8x^2 - 76x + 30

NEXT STEPS

Final Takeaways from Your Teacher

Practice distributing negative signs across multi-term expressions by drawing arrows to every term.
Whenever a minus sign sits in front of parentheses, mentally replace it with $-1 \cdot (\dots)$ so each term is multiplied by $-1$ .

OverviewACEpath Teacher's Diagnosis

03 The student incorrectly multiplies the terms of (x - 3)(5 - x) by treating the second factor as (x - 5), resulting in x² - 8x + 15 instead of -x² + 8x - 15.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
-3x[2x - 4x + 20] - 2(x^2 - 8x + 15)
Correct
-3x[2x - 4x + 20] - 2(-x^2 + 8x - 15)
You correctly distributed the $-4$ inside the brackets to get $2x - 4x + 20$ . However, when expanding $(x - 3)(5 - x)$ , you treated $(5 - x)$ as $(x - 5)$ , obtaining $x^2 - 8x + 15$ instead of $5x - x^2 - 15 + 3x = -x^2 + 8x - 15$ .
Step 2
-3x[-2x + 20] - 2x^2 + 16x - 30 $\implies$ 6x^2 - 60x - 2x^2 + 16x - 30
Correct
-3x[-2x + 20] - 2(-x^2 + 8x - 15) $\implies$ 6x^2 - 60x + 2x^2 - 16x + 30
You correctly simplified the bracket to $-2x + 20$ and distributed $-3x$ and $-2$ into their respective terms, but the second part carries over the sign error from Step 1.
Step 3
4x^2 - 44x - 30
Correct
8x^2 - 76x + 30
You correctly combined the like terms from your previous line, but the result differs from the standard form answer due to the earlier binomial expansion mistake.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly distributed the negative factor $-4$ across $(x - 5)$ to get $-4x + 20$ .

You successfully simplified inside grouping symbols before clearing outer brackets.

You correctly distributed $-3x$ across $(-2x + 20)$ to yield $6x^2 - 60x$ .

You combined like terms accurately according to their signs and degrees in the final step.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign Error in Non-Standard Binomial Expansion

You expanded $(x - 3)(5 - x)$ into $x^2 - 8x + 15$ by assuming the second term was standard form $(x - 5)$ rather than tracking the negative sign on $-x$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking non-standard term ordering in binomials (such as $c - x$ ) causes automatic sign attribution to default to $(x - c)$ , flipping the sign of every product involving the variable.

REASONING PATH

Your ACEpath teacher’s suggested approach

Standardize or Track Variable Order: In any binomial of the form $(c - x)$ , either rewrite it explicitly as $(-x + c)$ or write out the individual FOIL products term-by-term.
Before multiplying $(x - 3)(5 - x)$ , write it as $(x - 3)(-x + 5)$ so the leading $-x$ term is immediately clear.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the binomial product $(x - 3)(5 - x) = x(5) + x(-x) - 3(5) - 3(-x) = -x^2 + 8x - 15$ .
Substitute both simplified parts: $-3x(-2x + 20) - 2(-x^2 + 8x - 15)$ .
Distribute the outer factors: $(6x^2 - 60x) + (2x^2 - 16x + 30)$ .
Combine like terms: $(6x^2 + 2x^2) + (-60x - 16x) + 30 = 8x^2 - 76x + 30$ .

Final answer

8x^2 - 76x + 30

NEXT STEPS

Final Takeaways from Your Teacher

When a factor has the variable subtracted, such as $(a - x)$ , rewrite it as $(-x + a)$ before expanding to prevent sign slips.
Practice expanding binomial pairs where terms are written in non-standard order.
Double-check each individual product term when multiplying binomials with mixed signs.

Lock it in

Question

What is the result of expanding $-c(x - d)$ using the distributive property?

Answer

$-cx + cd$ . Multiplying the negative factor $-c$ by the negative term $-d$ produces a positive product $+cd$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student fails to distribute the negative sign to the second term inside the parentheses, writing -4(x - 5) as -4x - 20 instead of -4x + 20.

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 expands the product 2(x - 3)(5 - x) but subtracts only the first term rather than distributing the negative sign across all terms.

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 multiplies the terms of (x - 3)(5 - x) by treating the second factor as (x - 5), resulting in x² - 8x + 15 instead of -x² + 8x - 15.

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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