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

1. TRANSLATE the problem information

• We need to find an expression equivalent to \((3\mathrm{x} + 2)^2 - (3\mathrm{x} - 2)^2\)
• This involves squaring binomials and subtracting

2. INFER the most efficient approach

• Key insight: This has the form \((\mathrm{a} + \mathrm{b})^2 - (\mathrm{a} - \mathrm{b})^2\) where \(\mathrm{a} = 3\mathrm{x}\) and \(\mathrm{b} = 2\)
• We can either use the pattern \((\mathrm{a} + \mathrm{b})^2 - (\mathrm{a} - \mathrm{b})^2 = 4\mathrm{ab}\) or expand directly
• The pattern method is faster, but both work

3. SIMPLIFY using the pattern method

• Apply \((\mathrm{a} + \mathrm{b})^2 - (\mathrm{a} - \mathrm{b})^2 = 4\mathrm{ab}\) with \(\mathrm{a} = 3\mathrm{x}\) and \(\mathrm{b} = 2\)
• Result = \(4(3\mathrm{x})(2) = 24\mathrm{x}\)

Alternative approach: Direct expansion

• \((3\mathrm{x} + 2)^2 = 9\mathrm{x}^2 + 12\mathrm{x} + 4\)
• \((3\mathrm{x} - 2)^2 = 9\mathrm{x}^2 - 12\mathrm{x} + 4\)
• Subtract: \((9\mathrm{x}^2 + 12\mathrm{x} + 4) - (9\mathrm{x}^2 - 12\mathrm{x} + 4) = 24\mathrm{x}\)

Answer: C (\(24\mathrm{x}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the negative sign during subtraction.

For example, they might write:

\((9\mathrm{x}^2 + 12\mathrm{x} + 4) - (9\mathrm{x}^2 - 12\mathrm{x} + 4) = 9\mathrm{x}^2 + 12\mathrm{x} + 4 - 9\mathrm{x}^2 - 12\mathrm{x} - 4 = 0\)

This incorrect sign handling leads them to get confused and potentially select Choice D (\(9\mathrm{x}^2 - 4\)) by incorrectly applying difference of squares to the original binomials.

Second Most Common Error:

Missing INFER insight: Students don't recognize the efficient pattern and get bogged down in lengthy calculations, making arithmetic mistakes along the way.

Without seeing the \((\mathrm{a} + \mathrm{b})^2 - (\mathrm{a} - \mathrm{b})^2 = 4\mathrm{ab}\) pattern, they may struggle with the expansion and select Choice B (\(18\mathrm{x}^2 + 8\)) from incomplete or incorrect algebraic work.

The Bottom Line:

This problem rewards pattern recognition. Students who spot the \((\mathrm{a} + \mathrm{b})^2 - (\mathrm{a} - \mathrm{b})^2\) structure can solve it in seconds, while those who don't may struggle with error-prone expansions.