1. **Problem:** Factor the quadratic expression $x^2 + 7x + 12$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$ where $a=1$, find two numbers that multiply to $c$ and add to $b$. 3. **Step:** Find two numbers that multiply to 12 and add to 7. These are 3 and 4. 4. **Step:** Write the factorization as $(x + 3)(x + 4)$. 5. **Verification:** Expand $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$. **Final answer:** $\boxed{(x + 3)(x + 4)}$