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:

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

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

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

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

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

$(x+1)^2$

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.
To be of some more help, you should start by factorising as much as you can, to see if any factors can be cancelled.