Which expression is equivalent to \((8\mathrm{x}^2 - 3\mathrm{x}) - (2\mathrm{x}^2 + 5\mathrm{x})\)?

1. INFER the solution strategy

• Recognize this as polynomial subtraction requiring two main steps:
  • First: distribute the negative sign
  • Second: combine like terms

2. SIMPLIFY by distributing the negative sign

• Starting expression: \((8x^2 - 3x) - (2x^2 + 5x)\)
• Distribute the negative to each term in the second parentheses:
  • The negative sign applies to both \(2x^2\) AND \(5x\)
  • Result: \(8x^2 - 3x - 2x^2 - 5x\)

3. SIMPLIFY by combining like terms

• Group terms with the same variable and exponent:
  • \(x^2\) terms: \(8x^2 - 2x^2 = 6x^2\)
  • \(x\) terms: \(-3x - 5x = -8x\)
• Final expression: \(6x^2 - 8x\)

Answer: (A) \(6x^2 - 8x\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly distribute the negative sign, getting \(8x^2 - 3x - 2x^2 + 5x\) instead of \(8x^2 - 3x - 2x^2 - 5x\). They treat the subtraction like addition, forgetting that subtracting \((2x^2 + 5x)\) means subtracting both terms.

When they combine like terms with this error: \(8x^2 - 2x^2 = 6x^2\) and \(-3x + 5x = +2x\), they get \(6x^2 + 2x\).

This may lead them to select Choice B \((6x^2 + 2x)\).

Second Most Common Error:

Poor SIMPLIFY reasoning: Students correctly distribute the negative sign but make calculation errors when combining the negative terms. They might calculate \(-3x - 5x\) as \(-2x\) instead of \(-8x\), possibly due to rushing or not being careful with negative number operations.

This leads to confusion about the final coefficient and may cause them to guess among the remaining choices.

The Bottom Line:

This problem tests careful execution of sign rules - students who methodically track negative signs through distribution and combination succeed, while those who rush or forget that subtraction affects all terms in the parentheses typically select incorrect answers.