Given the following problem:

$\left(\frac{x^2-2x+1}{x^2+x}.\frac{2x+1}{x^2-x}\right)^{-1}+\frac{x^3-x}{(x+1)^2-x^2}$

$= \left(\frac{(x-1)^2}{x(x+1)}.\frac{2x+1}{x(x-1)}\right)^{-1}+\frac{x(x+1)(x-1)}{(x+1-x)(x+1+x)}$

$= \left(x-1\right)^{-1}+\frac{x(x+1)(x-1)}{(x+1-x)(x+1+x)}$

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

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

$(x+1)^2$

I am not sure if that is correct, and would love someone to review my process, as I tried to be as detailed as possible.

Sorry, but your process is VERY wrong.
You are evidently unaware of the basics: you treat, as example, (a + b) / a as being equal to b.
And x(x + 1) as being equal to (x + 1)^2.

You need serious classroom help. This site isn't a classroom...

Originally Posted by kkm
Given the following problem:

$\left(\frac{x^2-2x+1}{x^2+x}.\frac{2x+1}{x^2-x}\right)^{-1}+\frac{x^3-x}{(x+1)^2-x^2}$

$= \left(\frac{(x-1)^2}{x(x+1)}.\frac{2x+1}{x(x-1)}\right)^{-1}+\frac{x(x+1)(x-1)}{(x+1-x)(x+1+x)}$

$= \left(x-1\right)^{-1}+\frac{x(x+1)(x-1)}{(x+1-x)(x+1+x)}$

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

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

$(x+1)^2$