\((2\mathrm{a} + 3\mathrm{b})(\mathrm{a} - 4\mathrm{b}) + 6\mathrm{b}(3\mathrm{a} - 2\mathrm{b})\)Which of the following is equivalent to the expressi...

1. INFER the solution strategy

• We have two parts to expand: \((2\mathrm{a} + 3\mathrm{b})(\mathrm{a} - 4\mathrm{b})\) and \(6\mathrm{b}(3\mathrm{a} - 2\mathrm{b})\)
• Strategy: Expand each part separately, then combine like terms

2. SIMPLIFY the first expression \((2\mathrm{a} + 3\mathrm{b})(\mathrm{a} - 4\mathrm{b})\)

• Use the distributive property (FOIL):
  • \(2\mathrm{a} \times \mathrm{a} = 2\mathrm{a}^2\)
  • \(2\mathrm{a} \times (-4\mathrm{b}) = -8\mathrm{ab}\)
  • \(3\mathrm{b} \times \mathrm{a} = 3\mathrm{ab}\)
  • \(3\mathrm{b} \times (-4\mathrm{b}) = -12\mathrm{b}^2\)
• Combine: \(2\mathrm{a}^2 - 8\mathrm{ab} + 3\mathrm{ab} - 12\mathrm{b}^2 = 2\mathrm{a}^2 - 5\mathrm{ab} - 12\mathrm{b}^2\)

3. SIMPLIFY the second expression \(6\mathrm{b}(3\mathrm{a} - 2\mathrm{b})\)

• Distribute \(6\mathrm{b}\) to each term:
  • \(6\mathrm{b} \times 3\mathrm{a} = 18\mathrm{ab}\)
  • \(6\mathrm{b} \times (-2\mathrm{b}) = -12\mathrm{b}^2\)
• Result: \(18\mathrm{ab} - 12\mathrm{b}^2\)

4. SIMPLIFY by combining both expanded expressions

• Add: \((2\mathrm{a}^2 - 5\mathrm{ab} - 12\mathrm{b}^2) + (18\mathrm{ab} - 12\mathrm{b}^2)\)
• INFER which are like terms:
  • \(\mathrm{a}^2\) terms: \(2\mathrm{a}^2\)
  • \(\mathrm{ab}\) terms: \(-5\mathrm{ab} + 18\mathrm{ab} = 13\mathrm{ab}\)
  • \(\mathrm{b}^2\) terms: \(-12\mathrm{b}^2 + (-12\mathrm{b}^2) = -24\mathrm{b}^2\)
• Final result: \(2\mathrm{a}^2 + 13\mathrm{ab} - 24\mathrm{b}^2\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Incomplete SIMPLIFY execution: Students correctly expand the first part \((2\mathrm{a} + 3\mathrm{b})(\mathrm{a} - 4\mathrm{b}) = 2\mathrm{a}^2 - 5\mathrm{ab} - 12\mathrm{b}^2\), but then forget to expand and add the second part \(6\mathrm{b}(3\mathrm{a} - 2\mathrm{b})\).

They see their partial result matches an answer choice and stop there.
This leads them to select Choice B \((2\mathrm{a}^2 - 5\mathrm{ab} - 12\mathrm{b}^2)\).

Second Most Common Error:

Poor SIMPLIFY with sign handling: Students make sign errors during expansion or when combining like terms, particularly with negative coefficients like \(-4\mathrm{b}\) or \(-2\mathrm{b}\).

Common mistakes include treating \((-4\mathrm{b}) \times (3\mathrm{b})\) as \(+12\mathrm{b}^2\) instead of \(-12\mathrm{b}^2\), or combining \(-12\mathrm{b}^2 + (-12\mathrm{b}^2)\) incorrectly.
This may lead them to select Choice C or D depending on where the sign errors occur.

The Bottom Line:

This problem tests sustained algebraic manipulation across multiple steps. Students must maintain accuracy through two separate expansions and a final combination step, making it easy to lose track or make computational errors along the way.