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