Answer
$-\dfrac{4}{x-1}$
Work Step by Step
The given expression, $
\left( \dfrac{x}{x+1}-\dfrac{x}{x-1} \right)\div\dfrac{x}{2x+2}
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{(x-1)x-(x+1)x}{(x+1)(x-1)}\cdot\dfrac{2x+2}{x}
\\\\=
\dfrac{x^2-x-(x^2+x)}{(x+1)(x-1)}\cdot\dfrac{2(x+1)}{x}
\\\\=
\dfrac{x^2-x-x^2-x}{(x+1)(x-1)}\cdot\dfrac{2(x+1)}{x}
\\\\=
\dfrac{-2x}{(x+1)(x-1)}\cdot\dfrac{2(x+1)}{x}
\\\\=
\dfrac{-2\cancel{x}}{(\cancel{x+1})(x-1)}\cdot\dfrac{2(\cancel{x+1})}{\cancel{x}}
\\\\=
\dfrac{-4}{x-1}
\\\\=
-\dfrac{4}{x-1}
.\end{array}
