#### Answer

$x=\left\{ -7-8i,-7+8i \right\}$

#### Work Step by Step

Taking the square root of both sides (the Square Root Property) and using $i=\sqrt{-1}$, the solutions to the given equation, $
(x+7)^2=-64
,$ are
\begin{array}{l}\require{cancel}
x+7=\pm\sqrt{-64}
\\\\
x+7=\pm\sqrt{-1}\cdot\sqrt{64}
\\\\
x+7=\pm i\cdot\sqrt{(8)^2}
\\\\
x+7=\pm i\cdot8
\\\\
x+7=\pm 8i
\\\\
x=-7\pm 8i
.\end{array}
Hence, $
x=\left\{ -7-8i,-7+8i \right\}
.$