Giải thích các bước giải:

a.Ta có:

$M=\dfrac{x^3}{x-1}-\dfrac{x^2}{x+1}-\dfrac1{x-1}+\dfrac1{x+1}$

$\to M=\dfrac{x^3-1}{x-1}-\dfrac{x^2-1}{x+1}$

$\to M=\dfrac{(x-1)(x^2+x+1)}{x-1}-\dfrac{(x-1)(x+1)}{x+1}$

$\to M=x^2+x+1-(x-1)$

$\to M=x^2+2$

b.Ta có:

$N=(\dfrac{x-y}{x+y}+\dfrac{x+y}{x-y})\cdot (\dfrac{x^2+y^2}{2xy}+1)\cdot \dfrac{xy}{x^2+y^2}$

$\to N=\dfrac{(x-y)^2+(x+y)^2}{(x+y)(x-y)}\cdot \dfrac{x^2+y^2+2xy}{2xy}\cdot \dfrac{xy}{x^2+y^2}$

$\to N=\dfrac{2(x^2+y^2)}{(x+y)(x-y)}\cdot \dfrac{(x+y)^2}{2xy}\cdot \dfrac{xy}{x^2+y^2}$

$\to N=\dfrac{x+y}{x-y}$