Answer
$x=-12$
Work Step by Step
The factored form of the given equation, $
\dfrac{64}{x^2-16}+1=\dfrac{2x}{x-4}
,$ is
\begin{array}{l}\require{cancel}
\dfrac{64}{(x+4)(x-4)}+1=\dfrac{2x}{x-4}
.\end{array}
Multiplying both sides by the $LCD=
(x+4)(x-4)
,$ then the solution to the given equation is
\begin{array}{l}\require{cancel}
1(64)+(x+4)(x-4)(1)=(x+4)(2x)
\\\\
64+x^2-16=2x^2+8x
\\\\
x^2-2x^2-8x+64-16=0
\\\\
-x^2-8x+48=0
\\\\
x^2+8x-48=0
\\\\
(x+12)(x-4)=0
\\\\
x=\{ -12,4 \}
.\end{array}
Upon checking, only $
x=-12
$ satisfies the original equation.
