1. **Stating the problem:** Factorize the quadratic expression $x^2 + 7x + 12$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$. 3. Here, $a=1$, $b=7$, and $c=12$. We need two numbers that multiply to $1 \times 12 = 12$ and add to $7$. 4. The numbers $3$ and $4$ satisfy this because $3 \times 4 = 12$ and $3 + 4 = 7$. 5. Rewrite the middle term using these numbers: $$x^2 + 3x + 4x + 12$$ 6. Group terms: $$(x^2 + 3x) + (4x + 12)$$ 7. Factor each group: $$x(x + 3) + 4(x + 3)$$ 8. Factor out the common binomial: $$(x + 3)(x + 4)$$ 9. **Final answer:** The factorization of $x^2 + 7x + 12$ is $$\boxed{(x + 3)(x + 4)}$$.