Context: We need the expression that has $x + 2b$ as a factor, where $b$ is a positive integer constant. Each choice has the form $3x^2 + cx + 18b$.

Answer: D) $3x^2 + 39x + 18b$

Why: If $x + 2b$ divides $3x^2 + cx + 18b$, the other factor is $3x + k$, and $(x + 2b)(3x + k) = 3x^2 + (k + 6b)x + 2bk$. Matching constant terms gives $2bk = 18b$, so $k = 9$; matching middle terms gives $c = 6b + 9$. For choice D, $39 = 6b + 9$ yields $b = 5$, a positive integer. Check: with $b = 5$, $3x^2 + 39x + 90 = 3(x + 3)(x + 10)$, and $x + 2b = x + 10$ is indeed a factor.

Traps:

• A) Requires $6b + 9 = 9$, so $b = 0$, which is not a positive integer.
• B) Requires $6b + 9 = 24$, so $b = 2.5$, not an integer.
• C) Requires $6b + 9 = 30$, so $b = 3.5$, not an integer.