Answer
The solutions are $x=-15$ or $x=-1$.
Work Step by Step
$ x^{2}+16x+15=0\qquad$..add $-15$ to both sides so we can complete the square on the left side.
$ x^{2}+16x=-15\qquad$..add $64$ to both sides to complete the square ($\displaystyle \frac{1}{2}(16)=8$, and $(8)^{2}=64$.)
$ x^{2}+16x+64=-15+64\qquad$...simplify by applying
the Perfect square formula ($(x+a)^{2}=x^{2}+2ax+a^{2}$) and adding like terms.
$(x+8)^{2}=49$
According to the general principle of square roots:
For any real number $k$ and any algebraic expression $x$ :
$\text{If }x^{2}=k,\text{ then }x=\sqrt{k}\text{ or }x=-\sqrt{k}$.
$ x+8=\pm\sqrt{49}\qquad$...add $-8$ to each side.
$ x+8-8=\pm\sqrt{49}-8\qquad$...simplify.
$x=-8\pm 7$
$x=-8+7$ or $t=-8-7$
$x=-1$ or $x=-15$
