1. **State the problem:** Simplify or factor the quadratic expression $x^2 + 7x + 12$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Apply to our problem:** Here, $a=1$, $b=7$, and $c=12$. We need two numbers that multiply to $1 \times 12 = 12$ and add to $7$. 4. **Find the numbers:** The numbers $3$ and $4$ satisfy this because $3 \times 4 = 12$ and $3 + 4 = 7$. 5. **Write the factorization:** $$x^2 + 7x + 12 = (x + 3)(x + 4)$$ 6. **Explain:** This means the quadratic can be expressed as the product of two binomials, which is useful for solving equations or analyzing the function. **Final answer:** $$(x + 3)(x + 4)$$