Which of the following expressions has a factor of \((2\mathrm{x} + 4)\)? 6x^2 + 10x + 6 6x^2 + 18x...

1. INFER the key relationship

• If \(\mathrm{(2x + 4)}\) is a factor of an expression, then that expression can be written as:
  \(\mathrm{(2x + 4) \times (some\ other\ expression)}\)
• This means I should be able to factor each choice and find \(\mathrm{(2x + 4)}\) as one of the factors

2. SIMPLIFY by factoring each choice systematically

Choice A: \(\mathrm{6x^2 + 10x + 6}\)

• Factor out the GCF of 2: \(\mathrm{2(3x^2 + 5x + 3)}\)
• Since \(\mathrm{(2x + 4) = 2(x + 2)}\), I need \(\mathrm{(x + 2)}\) to be a factor of \(\mathrm{3x^2 + 5x + 3}\)
• Trying to factor \(\mathrm{3x^2 + 5x + 3}\): This doesn't factor nicely with \(\mathrm{(x + 2)}\)

Choice B: \(\mathrm{6x^2 + 18x + 12}\)

• Factor out the GCF of 6: \(\mathrm{6(x^2 + 3x + 2)}\)
• Factor the simpler quadratic: \(\mathrm{x^2 + 3x + 2 = (x + 1)(x + 2)}\)
• So: \(\mathrm{6x^2 + 18x + 12 = 6(x + 1)(x + 2) = 2(x + 2) \times 3(x + 1) = (2x + 4)(3x + 3)}\) ✓

3. SIMPLIFY to verify the result

• Check: \(\mathrm{(2x + 4)(3x + 3)}\)
  \(\mathrm{= 6x^2 + 6x + 12x + 12}\)
  \(\mathrm{= 6x^2 + 18x + 12}\) ✓
• This matches Choice B exactly!

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that finding a factor means the expression must be factorable in a specific way. Instead, students might try to divide each expression by \(\mathrm{(2x + 4)}\) using polynomial long division, which is much more complex and error-prone than factoring.

This leads to confusion and computational errors, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Making errors while factoring, such as incorrectly factoring quadratics or making arithmetic mistakes when finding factors. For example, incorrectly factoring \(\mathrm{x^2 + 3x + 2}\) as \(\mathrm{(x + 2)(x + 2)}\) instead of \(\mathrm{(x + 1)(x + 2)}\).

This may lead them to select Choice A or get confused between multiple choices.

The Bottom Line:

This problem tests whether students can connect the concept of "having a factor" with the concrete skill of factoring expressions. The key insight is recognizing that factoring is often easier than division for identifying factors.