Which expression is equivalent to \((4\mathrm{x}^2 - 3\mathrm{x} + 7) - (2\mathrm{x}^2 + 5\mathrm{x} - 2)\)?

1. TRANSLATE the problem information

• Given: \((4\mathrm{x}^2 - 3\mathrm{x} + 7) - (2\mathrm{x}^2 + 5\mathrm{x} - 2)\)
• Need to find: An equivalent expression (same value for any x-value)
• Key insight: This is polynomial subtraction requiring careful attention to signs

2. SIMPLIFY by distributing the negative sign

• The subtraction means we multiply each term in the second parentheses by -1:
  • \(2\mathrm{x}^2\) becomes \(-2\mathrm{x}^2\)
  • \(+5\mathrm{x}\) becomes \(-5\mathrm{x}\)
  • \(-2\) becomes \(+2\)
• Result: \(4\mathrm{x}^2 - 3\mathrm{x} + 7 - 2\mathrm{x}^2 - 5\mathrm{x} + 2\)

3. SIMPLIFY by combining like terms

• Group terms with same variable and power:
  • x² terms: \(4\mathrm{x}^2 - 2\mathrm{x}^2 = 2\mathrm{x}^2\)
  • x terms: \(-3\mathrm{x} - 5\mathrm{x} = -8\mathrm{x}\)
  • Constant terms: \(7 + 2 = 9\)
• Final expression: \(2\mathrm{x}^2 - 8\mathrm{x} + 9\)

Answer: D (\(2\mathrm{x}^2 - 8\mathrm{x} + 9\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Not properly distributing the negative sign through the second parentheses

Students often write: \(4\mathrm{x}^2 - 3\mathrm{x} + 7 - 2\mathrm{x}^2 + 5\mathrm{x} - 2\) (keeping the \(+5\mathrm{x}\) instead of changing it to \(-5\mathrm{x}\))

This leads to: \(2\mathrm{x}^2 + 2\mathrm{x} + 5\)

This may lead them to select Choice A (\(2\mathrm{x}^2 + 2\mathrm{x} + 5\))

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when combining like terms, even after correctly distributing

Students might correctly get to \(4\mathrm{x}^2 - 3\mathrm{x} + 7 - 2\mathrm{x}^2 - 5\mathrm{x} + 2\), but then make arithmetic errors like:

• Combining \(-3\mathrm{x} - 5\mathrm{x}\) incorrectly
• Getting confused with positive and negative terms

This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem tests careful attention to signs during algebraic manipulation. The key challenge is systematically applying the distributive property with negative signs, then methodically combining like terms without losing track of positive and negative values.