Let \(\mathrm{g(x) = (2x - 1)^2 - (2x - 1) - 12}\). Which of the following is a factor of...

1. INFER the best approach

• Looking at \(\mathrm{g(x) = (2x - 1)^2 - (2x - 1) - 12}\), I notice the expression \(\mathrm{(2x - 1)}\) appears multiple times
• This repetitive structure suggests using substitution to simplify the problem
• Strategy: Let \(\mathrm{y = 2x - 1}\) to transform this into a simpler quadratic

2. TRANSLATE using substitution

• Let \(\mathrm{y = 2x - 1}\)
• Then \(\mathrm{g(x)}\) becomes: \(\mathrm{y^2 - y - 12}\)
• Now I have a standard quadratic to factor instead of a complex expression

3. SIMPLIFY by factoring the quadratic

• Need to factor \(\mathrm{y^2 - y - 12}\)
• Looking for two numbers that multiply to -12 and add to -1
• Testing factor pairs of -12: \(\mathrm{(3)(-4) = -12}\) and \(\mathrm{3 + (-4) = -1}\) ✓
• So \(\mathrm{y^2 - y - 12 = (y - 4)(y + 3)}\)

4. TRANSLATE back to x terms

• \(\mathrm{y - 4 = (2x - 1) - 4 = 2x - 5}\)
• \(\mathrm{y + 3 = (2x - 1) + 3 = 2x + 2 = 2(x + 1)}\)
• Therefore: \(\mathrm{g(x) = (2x - 5) \cdot 2(x + 1)}\)

5. INFER the answer from the factor form

• From \(\mathrm{g(x) = 2(2x - 5)(x + 1)}\), I can see that \(\mathrm{(2x - 5)}\) is a factor
• Checking the answer choices, \(\mathrm{(2x - 5)}\) appears as choice (B)

Answer: B (2x - 5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to expand \(\mathrm{(2x - 1)^2 - (2x - 1) - 12}\) directly instead of recognizing the substitution opportunity. They get bogged down in complex algebraic manipulation:

• \(\mathrm{(2x - 1)^2 = 4x^2 - 4x + 1}\)
• Full expansion: \(\mathrm{4x^2 - 4x + 1 - 2x + 1 - 12 = 4x^2 - 6x - 10}\)

While this approach can work, students often make computational errors during the expansion or struggle to factor the resulting expression \(\mathrm{4x^2 - 6x - 10}\). This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize they need to factor \(\mathrm{y^2 - y - 12}\) after substitution, but incorrectly identify the factor pair. They might think the factors are \(\mathrm{(y - 3)(y + 4)}\) because they mixed up signs or calculation. When they substitute back, they get incorrect factors like \(\mathrm{(2x - 4)}\) or \(\mathrm{(2x + 7)}\), leading them to select Choice A \(\mathrm{(x - 4)}\) by incorrectly thinking \(\mathrm{x - 4}\) relates to their wrong factor \(\mathrm{2x - 4}\).

The Bottom Line:

This problem tests whether students can recognize when substitution dramatically simplifies a complex-looking expression. The key insight is seeing the repeated \(\mathrm{(2x - 1)}\) pattern and treating it as a single unit rather than expanding everything from the start.