(x+1)/(2x-2)-(x-1)/(2x+2)-(4x)/(x^2-1)+(x^2+1)/(x^2-1)=(x-1)/(x+1) why?

I made an error is my first post so I have done it again

$$\\\frac{(x+1)}{(2x-2)}-\frac{(x-1)}{(2x+2)}-\frac{(4x)}{(x^2-1)}+\frac{(x^2+1)}{(x^2-1)}=\frac{x-1}{x+1}$$

$$\\LHS=\frac{(x+1)}{2(x-1)}-\frac{(x-1)}{2(x+1)}+\frac{(x^2+1)}{(x^2-1)}-\frac{(4x)}{(x^2-1)}\\\\ LHS=\frac{(x+1)^2}{2(x-1)(x+1)}-\frac{(x-1)^2}{2(x+1)(x-1)}+\frac{(x^2+1)-4x}{(x^2-1)}\\\\ LHS=\frac{(x+1)^2-(x-1)^2}{2(x-1)(x+1)}+\frac{x^2+1-4x}{(x-1)(x+1)}\\\\ LHS=\frac{(x^2+2x+1)-(x^2-2x+1)}{2(x-1)(x+1)}+\frac{x^2-4x+1}{(x-1)(x+1)}\\\\ LHS=\frac{4x}{2(x-1)(x+1)}+\frac{x^2-4x+1}{(x-1)(x+1)}\\\\ LHS=\frac{2x}{(x-1)(x+1)}+\frac{x^2-4x+1}{(x-1)(x+1)}\\\\ LHS=\frac{2x+x^2-4x+1}{(x-1)(x+1)}\\\\ LHS=\frac{x^2-2x+1}{(x-1)(x+1)}\\\\ LHS=\frac{(x-1)(x-1)}{(x-1)(x+1)}\\\\ LHS=\frac{x-1}{x+1}\\\\ LHS =RHS$$