1. Let's clarify the factorization of the quadratic expression $x^2 + 7x + 12$. 2. The goal is to find two numbers that multiply to the constant term 12 and add up to the coefficient of $x$, which is 7. 3. The two numbers are 3 and 4 because: - $3 \times 4 = 12$ - $3 + 4 = 7$ 4. To show the multiplication explicitly: - $3 \times 4 = 12$ 5. The factorization is then written as: $$x^2 + 7x + 12 = (x + 3)(x + 4)$$ 6. This means when you expand $(x + 3)(x + 4)$, you get back the original quadratic: $$\begin{aligned} (x + 3)(x + 4) &= x \times x + x \times 4 + 3 \times x + 3 \times 4 \\ &= x^2 + 4x + 3x + 12 \\ &= x^2 + 7x + 12 \end{aligned}$$ 7. So, 3 and 4 multiply to 12 and add to 7, which is why the factorization works.