Power Series Representation of a Function, (differentiation)

Question

Suppose that $f(x) =\frac{x^2+x}{(1-x)^3}$. What would the power series representation for this function? I have found the power series representation for $\frac{1}{(1-x)^3}$ which is $\sum_{n=2}^{\infty} \frac{n(n-1)x^{n-2}}{2}$. Would it be allowed to multiply the series by $x^2+x$ to get $\left(x^2+x\right) \sum_{n=2}^{\infty} \frac{n(n-1)x^{n-2}}{2}$?

Answer 1

This is indeed correct and the result is then given by $\sum_{n=0}^{\infty}n^2x^n$, as stated in M. Iwaniuk's comment.

$$\left(x^2+x\right)\sum_{n=2}^{\infty}\frac{n(n-1)x^{n-2}}{2}=x^2\sum_{n=2}^{\infty}\frac{n(n-1)x^{n-2}}{2}+x\sum_{n=2}^{\infty}\frac{n(n-1)x^{n-2}}{2}=\sum_{n=2}^{\infty}\frac{n(n-1)x^{n}}{2}+\sum_{n=2}^{\infty}\frac{n(n-1)x^{n-1}}{2}=\sum_{n=2}^{\infty}\frac{n(n-1)x^{n}}{2}+\sum_{n=1}^{\infty}\frac{(n+1)nx^{n}}{2}=x+\sum_{n=2}^{\infty}\frac{n(n-1)+n(n+1)}{2}x^{n}=\sum_{n=0}^{\infty}n^2x^n$$, as $0x^0=0$ and $x=1^2x^1$