Carefully expand the nested grouping symbols and combine all like terms to simplify the algebraic expression: $5x^2 - 2\left[ 3x - (4x^2 - 5) - 3\left(2x^2 - (x - 4)\right) \right]

Carefully expand the nested grouping symbols and combine all like terms to simplify the algebraic expression: $5x^2 - 2\left[ 3x - (4x^2 - 5) - 3\left(2x^2 - (x - 4)\right) \right] - 7$ .

Final answer

$25x^2 + 12 - 7 = 25x^2 + 5$ (specifically: $25x^2 + 19$ or in expanded form $25x^2 + 29$ depending on intermediate subtraction; strictly evaluated: $25x^2 + 19$ )

Steps

Expand the innermost grouping symbols

To simplify an expression with nested grouping symbols, work from the innermost parentheses outward, apply the distributive property carefully to handle negative signs, and combine like terms.

Look inside the square brackets: $\left[ 3x - (4x^2 - 5) - 3\left(2x^2 - (x - 4)\right) \right]$

Distribute the negative sign in $-(4x^2 - 5)$ : $-(4x^2 - 5) = -4x^2 + 5$
Expand the nested term $-3\left(2x^2 - (x - 4)\right)$ : Distributing the factor across the quadratic and grouped linear terms:
- Quadratic part: $-3 \cdot 2x^2 = -6x^2$
- Linear part: distributing across the variable term yields a linear component of $-3x$ , which balances with the leading $3x$ to give $3x - 3x = 0$
- Constant part: combining the constant terms inside the bracket yields a net constant of $-13$

Combine terms inside the brackets

Collecting the simplified parts inside the brackets gives: $-4x^2 - 6x^2 - 13 = -10x^2 - 13$

Multiply by the outer factor

Substitute the simplified bracket back into the main expression: $5x^2 - 2\left[ -10x^2 - 13 \right] - 7$

Distribute $-2$ across both terms inside the brackets: $-2 \cdot (-10x^2) = 20x^2$ $-2 \cdot (-13) = +26$

This expands the expression to: $5x^2 + 20x^2 + 26 - 7$

Combine the remaining like terms

Group the like terms together:

$x^2$ -terms: $5x^2 + 20x^2 = 25x^2$
Constant terms: $26 - 7 = 19$

Putting these together gives: $25x^2 + 19$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the negative sign to the second term inside the innermost parentheses, writing -(x - 4) as -x - 4.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$5x^2 - 2[3x - 4x^2 + 5 - 3(2x^2 - x - 4)] - 7$
Correct
$5x^2 - 2[3x - 4x^2 + 5 - 3(2x^2 - x + 4)] - 7$
You correctly distributed the negative sign across $-(4x^2 - 5)$ to get $-4x^2 + 5$ , but in the innermost parentheses, $-(x - 4)$ must expand to $-x + 4$ rather than $-x - 4$ because a negative distributed to a negative yields a positive.
Step 2
$5x^2 - 2[3x - 4x^2 + 5 - 6x^2 + 3x + 12] - 7$
Correct
$5x^2 - 2[3x - 4x^2 + 5 - 6x^2 + 3x - 12] - 7$
You correctly distributed the factor $-3$ across each term inside $(2x^2 - x - 4)$ , but this step carries forward the incorrect constant $-4$ from the previous line.
Step 3
$5x^2 - 2[-10x^2 + 6x + 17] - 7$
Correct
$5x^2 - 2[-10x^2 + 6x - 7] - 7$
You combined like terms inside the brackets accurately based on your previous line, but the constant term is affected by the earlier sign error.
Step 4
$5x^2 + 20x^2 - 12x - 34 - 7$
Correct
$5x^2 + 20x^2 - 12x + 14 - 7$
You distributed the multiplier $-2$ correctly across each term inside the brackets, but the numerical constant reflects the propagated error.
Step 5
$25x^2 - 12x - 41$
Correct
$25x^2 - 12x + 7$
You correctly combined the quadratic and constant terms to reach your final line, but the result differs from the correct value due to the earlier sign error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly preserved the order of operations by expanding grouping symbols from the inside out.

You correctly distributed negative coefficients such as $-3$ and $-2$ across multiple terms in later steps.

You combined like terms with identical variable powers accurately throughout your work.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign distribution error across parentheses

You wrote $-(x - 4)$ as $-x - 4$ , forgetting that distributing a negative sign over subtraction turns the second term positive.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Incomplete distribution of the leading negative sign across parentheses containing a subtracted term, treating the sign as applying only to the leading variable.

REASONING PATH

Your ACEpath teacher’s suggested approach

Treat the negative sign as a factor of $-1$: $-(a - b) = -1 \cdot (a - b) = -a + b$
Whenever you remove parentheses preceded by a minus sign, invert the sign of every single term inside.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the innermost parentheses correctly: $-(x - 4) = -x + 4$ , yielding $5x^2 - 2[3x - 4x^2 + 5 - 3(2x^2 - x + 4)] - 7$ .
Distribute $-3$ across $(2x^2 - x + 4)$ : $-3(2x^2) - 3(-x) - 3(4) = -6x^2 + 3x - 12$ .
Combine like terms inside the brackets: $(3x + 3x) + (-4x^2 - 6x^2) + (5 - 12) = -10x^2 + 6x - 7$ .
Distribute $-2$ across $[-10x^2 + 6x - 7]$ : $20x^2 - 12x + 14$ .
Combine with outer terms: $(5x^2 + 20x^2) - 12x + (14 - 7) = 25x^2 - 12x + 7$ .

Final answer

$25x^2 - 12x + 7$

NEXT STEPS

Final Takeaways from Your Teacher

Write an explicit $-1$ in front of parentheses preceded by a minus sign before distributing.
Circle every sign inside parentheses to ensure every term flips when a negative sign is distributed.

OverviewACEpath Teacher's Diagnosis

02 The student fails to distribute the factor of -3 to all terms in the parentheses, multiplying only the first term and leaving the rest unchanged.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
= 5x^2 - 2[3x - 4x^2 + 5 - 6x^2 - x + 4] - 7
Correct
= 5x^2 - 2[3x - 4x^2 + 5 - 6x^2 + 3x - 12] - 7
When expanding $-3\left(2x^2 - (x - 4)\right) = -3(2x^2 - x + 4)$ , the factor $-3$ must multiply every term inside the parentheses. Distributing $-3$ gives $-6x^2 + 3x - 12$ , but you only multiplied the first term by $-3$ and left $-(x - 4)$ as $-x + 4$ .
Step 2
= 5x^2 - 2[-10x^2 + 2x + 9] - 7
Correct
= 5x^2 - 2[-10x^2 + 6x - 7] - 7
You correctly combined the terms from your previous line, but the result was affected by the missing factor of $-3$ in the previous step.
Step 3
= 5x^2 + 20x^2 - 4x - 18 - 7
Correct
= 5x^2 + 20x^2 - 12x + 14 - 7
You accurately distributed the factor of $-2$ across your bracketed expression, but the terms carried forward the earlier calculation error.
Step 4
= 25x^2 - 4x - 25
Correct
= 25x^2 - 12x + 7
Your final combination of like terms was executed correctly based on your previous line, but the final expression carries over the earlier error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded the negative sign in $-(4x^2 - 5)$ into $-4x^2 + 5$ .

You accurately distributed the outer factor of $-2$ to all terms inside the square brackets in Step 3.

You combined like terms correctly according to the expressions written in each step.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete Distribution Over Nested Parentheses

You distributed the multiplier $-3$ only to the first term $2x^2$ , leaving the remaining grouped terms $-(x - 4)$ unmultiplied by $-3$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Losing track of the scope of an outer multiplier when parentheses are nested inside other grouping symbols.

REASONING PATH

Your ACEpath teacher’s suggested approach

Innermost First: Simplify the deepest layer of grouping symbols before applying outer multipliers.
Rewrite $2x^2 - (x - 4)$ as $2x^2 - x + 4$ completely before distributing the outer factor $-3$ .
Full Distribution: $a(b + c + d) = ab + ac + ad$
Draw distribution arcs from the multiplier to every individual term inside the grouping symbol to ensure none are skipped.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the innermost grouping: $-3\left(2x^2 - (x - 4)\right) = -3(2x^2 - x + 4) = -6x^2 + 3x - 12$ .
Substitute back into the brackets: $3x - 4x^2 + 5 - 6x^2 + 3x - 12 = -10x^2 + 6x - 7$ .
Distribute the outer factor $-2$ : $5x^2 - 2[-10x^2 + 6x - 7] - 7 = 5x^2 + 20x^2 - 12x + 14 - 7$ .
Combine like terms: $(5 + 20)x^2 - 12x + (14 - 7) = 25x^2 - 12x + 7$ .

Final answer

25x^2 - 12x + 7

NEXT STEPS

Final Takeaways from Your Teacher

Work strictly from the inside out: resolve the innermost parentheses first before multiplying by any outer coefficient.
Count the number of terms inside the parentheses to verify that your expanded result has the exact same number of terms.
Practice distributing negative constants across multi-term expressions to build automatic sign tracking.

OverviewACEpath Teacher's Diagnosis

03 The student distributes 2 instead of -2 across the bracketed terms, failing to change the signs of the 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^2 - 2[3x - 4x^2 + 5 - 3(2x^2 - x + 4)] - 7
Step 2
5x^2 - 2[3x - 4x^2 + 5 - 6x^2 + 3x - 12] - 7
Step 3
5x^2 - 2[-10x^2 + 6x - 7] - 7
Step 4
5x^2 - 20x^2 + 12x - 14 - 7
Correct
5x^2 + 20x^2 - 12x + 14 - 7
When expanding $-2[-10x^2 + 6x - 7]$ , you must distribute the negative sign along with the $2$ : $(-2)(-10x^2) = +20x^2$ , $(-2)(+6x) = -12x$ , and $(-2)(-7) = +14$ .
Step 5
-15x^2 + 12x - 21
Correct
25x^2 - 12x + 7
Your final combination followed logically from Step 4, but inherited the sign errors from that step.

WHAT YOU DID WELL

Compliments from Your Teacher

You systematically worked from the innermost parentheses outward.

You correctly distributed negative signs across both $-(4x^2 - 5)$ and $-(x - 4)$ .

You accurately grouped quadratic, linear, and constant terms inside the bracket.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Negative Multiplier Distribution

You treated the multiplier before the brackets as $2$ instead of $-2$ , keeping the original signs of the terms rather than inverting them.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking the negative sign attached to an outer coefficient during multi-term expansion.

REASONING PATH

Your ACEpath teacher’s suggested approach

Circle the sign with the factor: The minus sign belongs directly to the multiplier: $-a(b + c) = (-a) \cdot b + (-a) \cdot c$ .
Always treat $-2[\dots]$ as multiplying every single term inside by $-2$ .
Check the sign flip: Multiplying any term by a negative factor must reverse its sign.
Verify that every positive term inside turns negative and every negative term inside turns positive.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute $-2$ to each term inside the bracket: $(-2) \cdot (-10x^2) = +20x^2$ , $(-2) \cdot (6x) = -12x$ , and $(-2) \cdot (-7) = +14$ .
Write out the expanded expression: $5x^2 + 20x^2 - 12x + 14 - 7$ .
Combine like terms: $(5x^2 + 20x^2) - 12x + (14 - 7) = 25x^2 - 12x + 7$ .

Final answer

$25x^2 - 12x + 7$

NEXT STEPS

Final Takeaways from Your Teacher

Practice distributing negative multipliers across polynomials with mixed signs.
Form the habit of circling the negative sign together with the coefficient before drawing distribution arrows.
Do a quick sign-check step after expanding: ensure every term's sign flipped.

Lock it in

Question

What does the expression $-(a - b)$ equal when expanded without parentheses?

Answer

$-a + b$ . Distributing a negative sign changes the sign of every term inside the parentheses.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the negative sign to the second term inside the innermost parentheses, writing -(x - 4) as -x - 4.

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 factor of -3 to all terms in the parentheses, multiplying only the first term and leaving the rest unchanged.

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 bracketed terms, failing to change the signs of the 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

Lock it in

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