#### Answer

$x=\{-7,-6\}$

#### Work Step by Step

Squaring both sides of the equation and then using the properties of equality, we obtain:
\begin{array}{l}\require{cancel}\left(
\sqrt{x+7}
\right)^2=\left(
x+7
\right)^2
\\\\
x+7=(x)^2+2(x)(7)+(7)^2
\\\\
x+7=x^2+14x+49
\\\\
0=x^2+(14x-x)+(49-7)
\\\\
x^2+13x+42=0
\\\\
(x+6)(x+7)=0
\\\\
x=\{-7,-6\}
.\end{array}
Upon checking, both solutions satisfy the original equation. Hence, $
x=\{-7,-6\}
.$