1. **State the problem:** We want to find for which values of $b$ the quadratic expression $$x^2 + bx + 12$$ is factorable over the integers. 2. **Recall the factorization rule:** A quadratic $x^2 + bx + c$ is factorable over integers if there exist integers $m$ and $n$ such that: $$m \times n = c$$ $$m + n = b$$ 3. **Apply to our problem:** Here, $c = 12$. We need integer pairs $(m,n)$ such that $m \times n = 12$. 4. **List factor pairs of 12:** $$(1, 12), (2, 6), (3, 4), (-1, -12), (-2, -6), (-3, -4)$$ 5. **Calculate sums for each pair:** - $1 + 12 = 13$ - $2 + 6 = 8$ - $3 + 4 = 7$ - $-1 + (-12) = -13$ - $-2 + (-6) = -8$ - $-3 + (-4) = -7$ 6. **Possible values of $b$ for factorability:** $$b \in \{7, 8, 13, -7, -8, -13\}$$ 7. **Match with options:** This corresponds to option D. **Final answer:** The expression is factorable for values of $b$ in option D: 7, 8, 13, -7, -8, -13.