Answer
$x=9$
Work Step by Step
The factored form of the given equation, $
\dfrac{36}{x^2-9}+1=\dfrac{2x}{x+3}
,$ is
\begin{array}{l}\require{cancel}
\dfrac{36}{(x+3)(x-3)}+1=\dfrac{2x}{x+3}
.\end{array}
Multiplying both sides by the $LCD=
(x+3)(x-3)
,$ then the solution to the given equation is
\begin{array}{l}\require{cancel}
1(36)+(x+3)(x-3)(1)=(x-3)(2x)
\\\\
36+x^2-9=2x^2-6x
\\\\
x^2-2x^2+6x+36-9=0
\\\\
-x^2+6x+27=0
\\\\
x^2-6x-27=0
\\\\
(x-9)(x+3)=0
\\\\
x=\{ -3,9 \}
.\end{array}
Upon checking, only $
x=9
$ satisfies the original equation.