Answer
$x=\left\{ -8,-6,0,2 .\right\}$
Work Step by Step
Let $z=
x^2+6x
.$ Then the given equation, $
x^2+6x=4\sqrt{x^2+6x}
$, is equivalent to
\begin{array}{l}\require{cancel}
z=4\sqrt{z}
.\end{array}
Squaring both sides of the equal sign results to
\begin{array}{l}\require{cancel}
z^2=16z
\\\\
z^2-16z=0
\\\\
z(z-16)=0
\\\\
z=\left\{ 0, 16 \right\}
.\end{array}
If $z=0$, then,
\begin{array}{l}\require{cancel}
x^2+6x=0
\\\\
x(x+6)=0
\\\\
x=\{ -6,0 \}
.\end{array}
If $z=16$, then,
\begin{array}{l}\require{cancel}
x^2+6x=16
\\\\
x^2+6x-16=0
\\\\
(x+8)(x-2)=0
\\\\
x=\{ -8,2 \}
.\end{array}
The proposed solutions are $
x=\left\{ -8,-6,0,2 .\right\}
$ Upon checking, all proposed solutions satisfy the original equation. Hence, $
x=\left\{ -8,-6,0,2 .\right\}
$
