`#3107.101107`
\(\dfrac{x+4y}{x^2-2xy}+\dfrac{x+y}{2y^2-xy}\)
\(=\dfrac{x+4y}{x\left(x-2y\right)}+\dfrac{x+y}{-y\left(x-2y\right)}\)
\(=\dfrac{y\left(x+4y\right)}{xy\left(x-2y\right)}-\dfrac{x\left(x+y\right)}{xy\left(x-2y\right)}\)
\(=\dfrac{xy+4y^2-x^2-xy}{xy\left(x-2y\right)}\)
\(=\dfrac{4y^2-x^2}{xy\left(x-2y\right)}\)
\(=\dfrac{\left(2y\right)^2-x^2}{xy\left(x-2y\right)}\)
\(=\dfrac{\left(2y-x\right)\left(2y+x\right)}{xy\left(x-2y\right)}\)
\(=\dfrac{-\left(2y+x\right)}{xy}=\dfrac{-2y-x}{xy}\)
\(\dfrac{x+4y}{x^2-2xy}+\dfrac{x+y}{2y^2-xy}\\ =\dfrac{x+4y}{x\left(x-2y\right)}+\dfrac{x+y}{y\left(2y-x\right)}\\ =\dfrac{x+4y}{x\left(x-2y\right)}-\dfrac{x+y}{y\left(x-2y\right)}\\ =\dfrac{y\left(x+4y\right)}{xy\left(x-2y\right)}-\dfrac{x\left(x+y\right)}{xy\left(x-2y\right)}\\ =\dfrac{xy+4y^2-x^2-xy}{xy\left(x-2y\right)}\\ =\dfrac{4y^2-x^2}{xy\left(x-2y\right)}\\ =\dfrac{-\left(x^2-4y^2\right)}{xy\left(x-2y\right)}\\ =\dfrac{-\left(x-2y\right)\left(x+2y\right)}{xy\left(x-2y\right)}\\ =\dfrac{-\left(x+2y\right)}{xy}\)
