Which of the following expressions is equivalent to \((9\mathrm{a}^2 - 2\mathrm{ab} + 5\mathrm{b}^2) - (4\mathrm{a}^2 + 3\mathrm{ab} - 3\mathrm{b}^2)\...

1. TRANSLATE the problem information

• Given: \((9\mathrm{a}^2 - 2\mathrm{ab} + 5\mathrm{b}^2) - (4\mathrm{a}^2 + 3\mathrm{ab} - 3\mathrm{b}^2)\)
• Need to find: The equivalent simplified expression

2. INFER the solution strategy

• When subtracting polynomials, we must distribute the negative sign to every term in the second polynomial
• After distribution, we can combine like terms by adding/subtracting their coefficients

3. SIMPLIFY by distributing the negative sign

• Original: \((9\mathrm{a}^2 - 2\mathrm{ab} + 5\mathrm{b}^2) - (4\mathrm{a}^2 + 3\mathrm{ab} - 3\mathrm{b}^2)\)
• Distribute the negative: \(9\mathrm{a}^2 - 2\mathrm{ab} + 5\mathrm{b}^2 - 4\mathrm{a}^2 - 3\mathrm{ab} + 3\mathrm{b}^2\)
• Notice: The \(-3\mathrm{b}^2\) becomes \(+3\mathrm{b}^2\) after distributing the negative

4. SIMPLIFY by grouping and combining like terms

• Group like terms: \((9\mathrm{a}^2 - 4\mathrm{a}^2) + (-2\mathrm{ab} - 3\mathrm{ab}) + (5\mathrm{b}^2 + 3\mathrm{b}^2)\)
• Combine coefficients:
  • \(\mathrm{a}^2\) terms: \(9 - 4 = 5\)
  • \(\mathrm{ab}\) terms: \(-2 - 3 = -5\)
  • \(\mathrm{b}^2\) terms: \(5 + 3 = 8\)
• Result: \(5\mathrm{a}^2 - 5\mathrm{ab} + 8\mathrm{b}^2\)

Answer: B (\(5\mathrm{a}^2 - 5\mathrm{ab} + 8\mathrm{b}^2\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students forget to distribute the negative sign to all terms in the second polynomial, particularly missing that \(-3\mathrm{b}^2\) becomes \(+3\mathrm{b}^2\).

For example, they might incorrectly write:
\(9\mathrm{a}^2 - 2\mathrm{ab} + 5\mathrm{b}^2 - 4\mathrm{a}^2 + 3\mathrm{ab} - 3\mathrm{b}^2\) (keeping \(+3\mathrm{ab}\) instead of \(-3\mathrm{ab}\))

This leads to: \(5\mathrm{a}^2 + \mathrm{ab} + 8\mathrm{b}^2\), causing them to select Choice C (\(5\mathrm{a}^2 + \mathrm{ab} + 8\mathrm{b}^2\)).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly distribute the negative sign but make arithmetic errors when combining like terms, especially with the \(\mathrm{b}^2\) terms.

They might calculate \(5\mathrm{b}^2 - 3\mathrm{b}^2 = 2\mathrm{b}^2\) instead of \(5\mathrm{b}^2 + 3\mathrm{b}^2 = 8\mathrm{b}^2\).

This leads them to select Choice A (\(5\mathrm{a}^2 - 5\mathrm{ab} + 2\mathrm{b}^2\)).

The Bottom Line:

This problem requires careful attention to sign changes when distributing and systematic organization when combining like terms. The key insight is that subtracting a polynomial means adding its opposite, which changes the sign of every term.