I presume you aren't allowed to use the quadratic formula. So I'll solve this using the "complete the square" method.

x^2 + 8x = -16

The form of a perfect square is

x^2 + (2a)x + a^2 = (x + a)^2

Comparing the LHS of both expressions (ie. comparing the coefficients of the x terms) I get that

2a = 8

a = 4

So we want to add a^2 = 4^2 = 16 to both sides of the equation. Thus

x^2 + 8x + 16 = -16 + 16

(x + 4)^2 = 0

x + 4 = 0

x = -4

-Dan