Answer
$\text{The simplified form of the given complex fraction is } \dfrac{x^{2} -x-8}{x^{2} +x-2}$.
Work Step by Step
$\begin{array}{ l l }
=\dfrac{( x( x-2))\left(\dfrac{x+4}{x} -\dfrac{3}{x-2}\right)}{( x( x-2))\left(\dfrac{x}{x-2} +\dfrac{1}{x}\right)} & \begin{array}{l}
\mathrm{Multiply\ the\ numerator\ and\ }\\
\mathrm{denominator\ of\ the\ complex\ }\\
\mathrm{fraction\ by\ its\ LCD}\\
x( x-2)
\end{array}\\
& \\
=\dfrac{( x( x-2))\dfrac{x+4}{x} -( x( x-2))\dfrac{3}{x-2}}{( x( x-2))\dfrac{x}{x-2} +( x( x-2))\dfrac{1}{x}} & \mathrm{Apply\ the\ distributive\ property}\\
& \\
=\dfrac{( x-2)( x+4) -3x}{x( x) +( x-2)( 1)} & \mathrm{Cancel\ factors\ of\ one}\\
& \\
=\dfrac{\left( x^{2} +2x-8\right) -3x}{x^{2} +x-2} & \begin{array}{l}
\mathrm{Expand} \ ( x-2)( x+4) \ \mathrm{using\ FOIL}\\
\mathrm{Multiply\ the\ all\ other\ terms}
\end{array}\\
& \\
=\dfrac{x^{2} +2x-3x-8}{x^{2} +x-2} & \begin{array}{l}
\mathrm{Group\ like\ terms\ together\ in\ the\ }\\
\mathrm{numerator}
\end{array}\\
& \\
=\dfrac{x^{2} -x-8}{x^{2} +x-2} & \mathrm{Combine\ like\ terms.}
\end{array}$
