\(\dfrac{4xy}{y^2-x^2}:\left(\dfrac{1}{y^2-x^2}+\dfrac{1}{x^2+2xy+y^2}\right)\)
\(=\dfrac{4xy}{\left(y-x\right)\left(x+y\right)}:\left(\dfrac{1}{\left(x+y\right)^2}-\dfrac{1}{\left(x-y\right)\left(x+y\right)}\right)\)
\(=\dfrac{4xy}{\left(y-x\right)\left(x+y\right)}:\left(\dfrac{\left(x-y\right)}{\left(x+y\right)^2\left(x-y\right)}-\dfrac{x+y}{\left(x-y\right)\left(x+y\right)^2}\right)\)
\(=\dfrac{4}{\left(y-x\right)\left(x+y\right)}:\dfrac{x-y-x-y}{\left(x+y\right)^2\left(x-y\right)}\)
\(=\dfrac{4}{\left(y-x\right)\left(x+y\right)}\cdot\dfrac{\left(x+y\right)^2\left(x-y\right)}{-2y}\)
\(=\dfrac{-4\left(x+y\right)^2\left(y-x\right)}{\left(y-x\right)\left(x+y\right)\left(-2y\right)}\)
\(=\dfrac{-4\left(x+y\right)}{-2y}\\ =\dfrac{2\left(x+y\right)}{y}\)
