Which expression is equivalent to \((4\mathrm{x}^2 - 3\mathrm{x} + 6) - (\mathrm{x}^2 + 5\mathrm{x} - 8)\)?

1. TRANSLATE the problem information

• We need to find an equivalent expression for \((4x^2 - 3x + 6) - (x^2 + 5x - 8)\)
• This means we're subtracting the entire second polynomial from the first polynomial

2. INFER the approach

• When subtracting polynomials, we must distribute the negative sign to every term in the second polynomial
• Then we can combine like terms to simplify

3. SIMPLIFY by distributing the negative sign

• \((4x^2 - 3x + 6) - (x^2 + 5x - 8)\)
• \(= 4x^2 - 3x + 6 - x^2 - 5x - (-8)\)
• \(= 4x^2 - 3x + 6 - x^2 - 5x + 8\)

4. SIMPLIFY by grouping like terms

• Group x² terms: \((4x^2 - x^2)\)
• Group x terms: \((-3x - 5x)\)
• Group constant terms: \((6 + 8)\)
• \(= (4x^2 - x^2) + (-3x - 5x) + (6 + 8)\)

5. SIMPLIFY by combining like terms

• \(4x^2 - x^2 = 3x^2\)
• \(-3x - 5x = -8x\)
• \(6 + 8 = 14\)
• \(= 3x^2 - 8x + 14\)

Answer: C) \(3x^2 - 8x + 14\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the negative sign must be distributed to every term in the second polynomial

Students often treat the subtraction like addition, simply combining corresponding terms without distributing the negative sign:
\((4x^2 - 3x + 6) + (x^2 + 5x - 8) = 5x^2 + 2x - 2\)

Or they might distribute the negative to only the first term: \(4x^2 - 3x + 6 - x^2 + 5x - 8 = 3x^2 + 2x - 2\)

This leads them to select Choice B (\(3x^2 - 8x - 2\)) or get confused about the signs.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when distributing or combining terms

Students might correctly understand they need to distribute the negative sign, but make calculation errors like:

• Incorrectly calculating \(-3x - 5x\) as \(-2x\) instead of \(-8x\)
• Getting confused with \(6 + 8\) vs \(6 - 8\) when handling the constant terms

This may lead them to select Choice A (\(3x^2 + 2x + 14\)) if they add the x terms instead of subtracting.

The Bottom Line:

Polynomial subtraction requires careful attention to signs - the negative sign affects every term being subtracted, not just the first one. Students must systematically distribute before combining.