\(\frac{2x}{x^2+2xy}+\frac{y}{xy-2y^2}+\frac{4}{x^2-4y^2}\\ \Leftrightarrow\frac{2x}{x\cdot\left(x+2y\right)}+\frac{y}{y\cdot\left(x-2y\right)}+\frac{4}{\left(x-2y\right)\cdot\left(x+2y\right)}\\ \Leftrightarrow\frac{2}{x+2y}+\frac{1}{x-2y}+\frac{4}{\left(x-2y\right)\cdot\left(x+2y\right)}\\ \Leftrightarrow\frac{2x-4y}{\left(x-2y\right)\cdot\left(x+2y\right)}+\frac{x+2y}{\left(x-2y\right)\cdot\left(x+2y\right)}+\frac{4}{\left(x-2y\right)\cdot\left(x+2y\right)}\\ \Leftrightarrow\frac{2x-4y+x+2y+4}{\left(x-2y\right)\cdot\left(x+2y\right)}\\ \Leftrightarrow\frac{3x-2y+4}{\left(x-2y\right)\cdot\left(x+2y\right)}\)