To complete the square for the given quadratic expression \( x^2 + 12x + 44 \), let's follow these steps:
1. Start with the original quadratic expression:
[tex]\[
x^2 + 12x + 44
\][/tex]
2. Isolate the quadratic and linear terms:
[tex]\[
x^2 + 12x
\][/tex]
3. To complete the square, take the coefficient of \( x \), which is 12, divide it by 2, and then square the result:
[tex]\[
\left(\frac{12}{2}\right)^2 = 6^2 = 36
\][/tex]
4. Add and subtract this squared value (36) inside the expression:
[tex]\[
x^2 + 12x + 36 - 36 + 44
\][/tex]
5. Group the perfect square trinomial and the constants separately:
[tex]\[
(x^2 + 12x + 36) + (-36 + 44)
\][/tex]
6. Rewrite the perfect square trinomial as a binomial square:
[tex]\[
(x + 6)^2 + (-36 + 44)
\][/tex]
7. Simplify the constants:
[tex]\[
-36 + 44 = 8
\][/tex]
8. So, the completed square form of the quadratic expression is:
[tex]\[
x^2 + 12x + 44 = (x + 6)^2 + 8
\][/tex]
Thus, filling in the gap in the equation, we have:
[tex]\[
x^2 + 12x + 44 = (x + 6)^2 + 8
\][/tex]
The value that completes the square is [tex]\( \boxed{8} \)[/tex].
