Question: Consider the quadratic equation $x^2 + 10x + 27 = 0$. Completing the square leads to the equivalent equation $(x + [ ] )^2 = [ ]$.
1. **State the problem:** We are given the quadratic equation $x^2 + 10x + 27 = 0$ and need to complete the square to rewrite it in the form $(x + a)^2 = b$.
2. **Recall the formula for completing the square:** For a quadratic $x^2 + bx + c = 0$, we can write it as $(x + \frac{b}{2})^2 = \left(\frac{b}{2}\right)^2 - c$.
3. **Apply the formula:** Here, $b = 10$ and $c = 27$.
Calculate $\frac{b}{2} = \frac{10}{2} = 5$.
Calculate $\left(\frac{b}{2}\right)^2 = 5^2 = 25$.
4. **Rewrite the equation:**
$$x^2 + 10x + 27 = 0 \implies (x + 5)^2 = 25 - 27$$
5. **Simplify the right side:**
$$25 - 27 = -2$$
6. **Final completed square form:**
$$(x + 5)^2 = -2$$
**Answer:** The values to fill in the blanks are $5$ and $-2$ respectively.