1. The problem is to complete the square for the polynomial $x^2 + 12x + \_$ to make it a perfect-square quadratic. 2. The formula to complete the square for $x^2 + bx$ is to add $\left(\frac{b}{2}\right)^2$. 3. Here, $b = 12$, so calculate $\left(\frac{12}{2}\right)^2 = 6^2 = 36$. 4. Adding 36 to the polynomial gives $x^2 + 12x + 36$, which factors as $\left(x + 6\right)^2$. 5. Therefore, the number to fill in the blank is 36. Final answer: 36