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

1. TRANSLATE the problem information

• Given: \((5\mathrm{x}^2 + 8\mathrm{x} - 3) - (2\mathrm{x}^2 - 4\mathrm{x} + 6)\)
• Need to find: The equivalent simplified expression

2. INFER the approach

• Strategy: Distribute the negative sign to remove parentheses, then combine like terms
• Key insight: The subtraction sign applies to every term in the second polynomial

3. SIMPLIFY by distributing the negative sign

Distribute the negative to each term in \((2\mathrm{x}^2 - 4\mathrm{x} + 6)\):

• \((5\mathrm{x}^2 + 8\mathrm{x} - 3) - (2\mathrm{x}^2 - 4\mathrm{x} + 6) = 5\mathrm{x}^2 + 8\mathrm{x} - 3 - 2\mathrm{x}^2 + 4\mathrm{x} - 6\)
• Notice: \(-(-4\mathrm{x})\) becomes \(+4\mathrm{x}\)

4. SIMPLIFY by grouping like terms

Rearrange terms by degree:

• \((5\mathrm{x}^2 - 2\mathrm{x}^2) + (8\mathrm{x} + 4\mathrm{x}) + (-3 - 6)\)

5. SIMPLIFY by combining like terms

• x² terms: \(5\mathrm{x}^2 - 2\mathrm{x}^2 = 3\mathrm{x}^2\)
• x terms: \(8\mathrm{x} + 4\mathrm{x} = 12\mathrm{x}\)
• Constant terms: \(-3 - 6 = -9\)

Answer: C. \(3\mathrm{x}^2 + 12\mathrm{x} - 9\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign error when distributing the negative sign

Students often forget that subtracting a negative becomes positive, writing:

\(5\mathrm{x}^2 + 8\mathrm{x} - 3 - 2\mathrm{x}^2 - 4\mathrm{x} - 6\) (incorrect!)

This leads to: \(3\mathrm{x}^2 + 4\mathrm{x} - 9\)

This may lead them to select Choice A (\(3\mathrm{x}^2 + 4\mathrm{x} - 9\))

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic mistakes when combining coefficients

Students correctly distribute signs but make calculation errors like:

• \(8\mathrm{x} + 4\mathrm{x} = 11\mathrm{x}\) (instead of \(12\mathrm{x}\))

This may lead them to select Choice B (\(3\mathrm{x}^2 + 11\mathrm{x} - 9\))

The Bottom Line:

This problem tests careful attention to signs and systematic organization. The key challenge is correctly handling the negative sign distribution while maintaining accuracy through multiple algebraic steps.