Let's solve the equation step-by-step to determine the missing number.
1. Start with the given equation:
[tex]\[
8x^2 + 80x = -5
\][/tex]
2. Factor out the 8 from the left-hand side:
[tex]\[
8(x^2 + 10x) = -5
\][/tex]
3. To complete the square inside the parentheses, we need to add and subtract a certain number. The goal is to turn [tex]\(x^2 + 10x\)[/tex] into a perfect square trinomial.
[tex]\[
x^2 + 10x + (constant)
\][/tex]
4. The constant can be found by taking half of the coefficient of [tex]\(x\)[/tex], which is 10, and then squaring it:
[tex]\[
\left(\frac{10}{2}\right)^2 = 5^2 = 25
\][/tex]
5. Add this constant inside the parentheses:
[tex]\[
8(x^2 + 10x + 25)
\][/tex]
6. However, we can't just add 25 inside the parentheses without making an equivalent adjustment on the right-hand side. Since we added [tex]\(25\)[/tex] inside the parentheses, which is being multiplied by 8, we need to add [tex]\(8 \times 25\)[/tex] to the right-hand side of the equation as well:
[tex]\[
8(x^2 + 10x + 25) = -5 + 8 \times 25
\][/tex]
7. Calculate [tex]\(8 \times 25\)[/tex]:
[tex]\[
8 \times 25 = 200
\][/tex]
Thus, the missing number is:
[tex]\[
200
\][/tex]
So the missing number in the last step is [tex]\(200\)[/tex].
