Answer
$x=\{ 4,20 \}$
Work Step by Step
Squaring both sides of the given equation, $
\sqrt{2x-4}-\sqrt{3x+4}=-2
,$ results to
\begin{array}{l}\require{cancel}
(\sqrt{2x-4}-\sqrt{3x+4})^2=(-2)^2
\\
(\sqrt{2x-4})^2-2(\sqrt{2x-4})(\sqrt{3x+4})+(\sqrt{3x+4})^2=4
\\
2x-4-2\sqrt{2x-4}\sqrt{3x+4}+3x+4=4
\\
5x-2\sqrt{2x-4}\sqrt{3x+4}=4
\\
5x-4=2\sqrt{2x-4}\sqrt{3x+4}
.\end{array}
Squaring both sides for the second time results to
\begin{array}{l}\require{cancel}
(5x-4)^2=(2\sqrt{2x-4}\sqrt{3x+4})^2
\\
(5x)^2-2(5x)(4)+(4)^2=4(2x-4)(3x+4)
\\
25x^2-40x+16=4(6x^2-4x-16)
\\
25x^2-40x+16=24x^2-16x-64
\\
(25x^2-24x^2)+(-40x+16x)+(16+64)=0
\\
x^2-24x+80=0
\\
(x-20)(x-4)=0
.\end{array}
Equating each factor to zero (Zero Product Property) and then solving for the variable, the solutions are $
x=\{ 4,20 \}
.$
Upon checking, both solutions, $
x=\{ 4,20 \}
,$ satisfy the original equation.