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

1. TRANSLATE the problem information

• Given: \((2\mathrm{x}^2 - 4) - (-3\mathrm{x}^2 + 2\mathrm{x} - 7)\)
• We need to subtract the second polynomial from the first

2. SIMPLIFY by distributing the negative sign

• When subtracting a polynomial, we distribute the negative to every term inside the second parenthesis
• \((2\mathrm{x}^2 - 4) - (-3\mathrm{x}^2 + 2\mathrm{x} - 7)\) becomes:
• \(2\mathrm{x}^2 - 4 + 3\mathrm{x}^2 - 2\mathrm{x} + 7\)
• Notice: \(-(-3\mathrm{x}^2) = +3\mathrm{x}^2\), \(-(+2\mathrm{x}) = -2\mathrm{x}\), \(-(-7) = +7\)

3. SIMPLIFY by combining like terms

• Group terms with the same variables and exponents:
  • \(\mathrm{x}^2\) terms: \(2\mathrm{x}^2 + 3\mathrm{x}^2 = 5\mathrm{x}^2\)
  • \(\mathrm{x}\) terms: \(-2\mathrm{x}\) (only one x term)
  • constants: \(-4 + 7 = 3\)
• Final expression: \(5\mathrm{x}^2 - 2\mathrm{x} + 3\)

Answer: A. \(5\mathrm{x}^2 - 2\mathrm{x} + 3\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students fail to properly distribute the negative sign when subtracting polynomials.

Instead of changing \(-(-3\mathrm{x}^2)\) to \(+3\mathrm{x}^2\), \(-(+2\mathrm{x})\) to \(-2\mathrm{x}\), and \(-(-7)\) to \(+7\), they might subtract each term as written: \(2\mathrm{x}^2 - 4 - 3\mathrm{x}^2 + 2\mathrm{x} - 7\). This gives them \(-\mathrm{x}^2 + 2\mathrm{x} - 11\).

This may lead them to select Choice D (\(-\mathrm{x}^2 + 2\mathrm{x} - 11\)).

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly distribute the negative but make sign errors when combining like terms.

They might get the distribution right (\(2\mathrm{x}^2 - 4 + 3\mathrm{x}^2 - 2\mathrm{x} + 7\)) but then incorrectly combine terms, such as treating \(-2\mathrm{x}\) as \(+2\mathrm{x}\) in the final answer.

This may lead them to select Choice B (\(5\mathrm{x}^2 + 2\mathrm{x} + 3\)) or similar incorrect combinations.

The Bottom Line:

This problem tests whether students can systematically apply the distributive property with negative signs and carefully track positive and negative terms throughout the simplification process. The key is remembering that subtracting a negative makes it positive.