In the answers I found on google, see this link, they converted the given function into a certain form? What is the process of that conversion (I understand it is a partial fraction of sorts, but how to get there I do not know)? And in the subsequent steps, the factorial part made no sense to me either.
So yes, how do I proceed?
welcome to MSE; As a hint:try like below $$\frac{x^4}{(x-1)(x-2)}=ax^2+bx+c+\frac{d}{x-1}+\frac{e}{x-2}$$ find $a,b,c,d,e$ then apply derivation. Remark: You can use dividing $x^4$ to $(x-1)(x-2)$ to simplify the fraction. It is not so hard to find $a,b,c,d,e$ for example if you think twice you will find $a=1$ and you can multiply both sides by $(x-1)$ and put $x=1 $ to find $d$ , and multiply bot sides by $(x-2)$ then put $x=2$ to find $e$
