a)
$\dfrac{x^2}{x+1} + \dfrac{2x}{(x-1)(x+1)} +\dfrac 1{1+x}+1$
$=\dfrac{x^2(x-1)+2x+(x-1)+(x-1)(x+1)}{(x-1)(x+1)}$
$=\dfrac{x^3-x^2+3x-1+x^2-1}{(x-1)(x+1)}$
$=\dfrac{x^3+3x-2}{(x-1)(x+1)}$
b) $\dfrac{2x+y}{2x^2-y} + \dfrac{8y}{y^2-4x^2}+\dfrac{2x-y}{2x^3+xy}$
$=\dfrac{2x+y}{2x^2-y}+\dfrac{8y}{(y-2x^2)(y+2x^2)}+\dfrac{2x-y}{x(2x^2+y)}$
$=\dfrac{-(2x+y)(y+2x^2)+8y+(2x-y)(y-2x^2)}{(y-2x^2)(y+2x^2)}$
$=\dfrac{-2xy-4x^3-y^2-2x^2y+8y+2xy-4x^3-y^2+2x^2y}{(y-2x^2)(y+2x^2)}$
$=\dfrac{-8x^3-2y^2+8y}{(y-2x^2)(y+2x^2)}$
c)
$\dfrac1{x-y} +\dfrac{3xy}{(y-x)(y^2+yx+x^2)} +\dfrac{ x-y}{x^2+xy+y^2}$
$=\dfrac{-(x^2+xy+y^2)+3xy+(x-y)(y-x)}{(y-x)(y^2+yx+x^2)}$
$=\dfrac{-x^2+2xy-y^2-(x^2-2xy+y^2)}{(y-x)(y^2+yx+x^2)}$
$=\dfrac{-2x^2+4xy-2y^2}{(y-x)(y^2+yx+x^2)}$
$=\dfrac{-2(x^2-2xy-y^2)}{(y-x)(y^2+yx+x^2)}$
$=\dfrac{-2(x-y)^2}{(y-x)(y^2+yx+x^2)}$
$=\dfrac{2(x-y)}{(y^2+yx+x^2)}$
