Answer
$x=-2\pm\sqrt{11}$
Work Step by Step
The factored form of the given equation, $
\dfrac{4}{x+2}+\dfrac{2x}{x-2}=\dfrac{6}{x^2-4}
,$ is
\begin{array}{l}\require{cancel}
\dfrac{4}{x+2}+\dfrac{2x}{x-2}=\dfrac{6}{(x+2)(x-2)}
.\end{array}
Multiplying both sides by the $LCD=
(x+2)(x-2)
,$ the solution/s of the given equation is/are
\begin{array}{l}\require{cancel}
(x-2)(4)+(x+2)(2x)=1(6)
\\\\
4x-8+2x^2+4x=6
\\\\
2x^2+(4x+4x)+(-8-6)=0
\\\\
2x^2+8x-14=0
\\\\
\dfrac{2x^2+8x-14}{2}=\dfrac{0}{2}
\\\\
x^2+4x-7=0
.\end{array}
Using $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ (or the Quadratic Formula), the solution/s of the above equation is/are
\begin{array}{l}\require{cancel}
x=\dfrac{-(4)\pm\sqrt{(4)^2-4(1)(-7)}}{2(1)}
\\\\
x=\dfrac{-4\pm\sqrt{16+28}}{2}
\\\\
x=\dfrac{-4\pm\sqrt{44}}{2}
\\\\
x=\dfrac{-4\pm\sqrt{4\cdot11}}{2}
\\\\
x=\dfrac{-4\pm\sqrt{(2)^2\cdot11}}{2}
\\\\
x=\dfrac{-4\pm2\sqrt{11}}{2}
\\\\
x=\dfrac{2(-2\pm\sqrt{11})}{2}
\\\\
x=\dfrac{\cancel{2}(-2\pm\sqrt{11})}{\cancel{2}}
\\\\
x=-2\pm\sqrt{11}
.\end{array}
