Answer
The solutions are $-4+\sqrt{15}$ and $-4-\sqrt{15}$.
Work Step by Step
$ x^{2}+8x=-1\qquad$ ...square half the coefficient of $x$.
$(\displaystyle \frac{8}{2})^{2}=4^{2}=16\qquad$ ...add $16$ to each side of the expression
$ x^{2}+8x+16=-1+16\qquad$ ... write left side as a binomial squared.
$(x+4)^{2}=-1+16\qquad$ ...simplify.
$(x+4)^{2}=15\qquad$ ...take square roots of each side.
$ x+4=\pm\sqrt{15}\qquad$ ...add $-4$ to each side
$ x+4-4=\pm\sqrt{15}-4\qquad$ ...simplify.
$x=-4\pm\sqrt{15}$
