Which of the following expressions has a factor of x + 2b, where b is a positive integer constant?

1. TRANSLATE the problem information

• Given: We need to find which expression has a factor of \((x + 2b)\) where \(b\) is a positive integer
• All choices have the form: \(3x² + [\mathrm{different\ middle\ terms}] + 14b\)

2. INFER the required factored form

• If \((x + 2b)\) is a factor, the expression must look like: \((x + 2b) \times (\mathrm{something})\)
• Since the first term is \(3x²\), the other factor must start with \(3x\)
• Since the constant term is \(14b\), and one factor contributes \(2b\), the other must contribute \(7\) to get \(14b\) total
• Therefore, the complete factorization must be: \((x + 2b)(3x + 7)\)

3. SIMPLIFY by expanding the factored form

• \((x + 2b)(3x + 7) = 3x² + 7x + 6bx + 14b\)
• Combine like terms: \(3x² + (7 + 6b)x + 14b\)
• This means the coefficient of \(x\) must equal \(7 + 6b\)

4. APPLY CONSTRAINTS to find the valid choice

• Test each choice by setting its middle coefficient equal to \(7 + 6b\):

• Choice A: \(7 = 7 + 6b \rightarrow b = 0\) (not positive ✗)
• Choice B: \(28 = 7 + 6b \rightarrow b = 3.5\) (not integer ✗)
• Choice C: \(42 = 7 + 6b \rightarrow b = 35/6\) (not integer ✗)
• Choice D: \(49 = 7 + 6b \rightarrow b = 7\) (positive integer ✓)

Answer: D. \(3x² + 49x + 14b\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that they need to work backwards from the desired factor to find the complete factorization. Instead, they might try to factor each expression individually without using the constraint that \((x + 2b)\) must be a factor. This leads to confusion since they can't easily factor expressions with unknown constants, causing them to abandon systematic solution and guess.

Second Most Common Error:

Missing APPLY CONSTRAINTS reasoning: Students correctly find the relationship \(7 + 6b = [\mathrm{middle\ coefficient}]\) and solve for \(b\) in each case, but forget to check that \(b\) must be a positive integer. They might select Choice B (\(3x² + 28x + 14b\)) because \(b = 3.5\) "looks reasonable" without recognizing that 3.5 isn't an integer.

The Bottom Line:

This problem requires recognizing that you must work backwards from the given factor constraint rather than trying to factor the expressions directly. The key insight is setting up the required factored form first, then using algebra to determine which choice satisfies the positive integer constraint on \(b\).