Prove generating function of $a_n = (b+1)^n - b^n$ The problem is prove that $$\frac{x}{(1-bx)(1-bx-x)}$$
is the generating function for the sequence $$a_n = (b+1)^n - b^n$$
I tried separating the fraction as $$x\frac{1}{1-bx}\frac{1}{1-x(b+1)}$$
which yields the generating functions $$x\sum_{n=0}^\infty (bx)^n\sum_{n=0}^\infty ((b+1)x)^n$$
Did I make a mistake? If not, how does this lead to the generating function with terms $a_n$?

HINT
$$
\begin{split}
\sum_{n=0}^\infty a_nx^n
 = \sum_{n=0}^\infty (b+1)^nx^n - \sum_{n=0}^\infty b^n x^n
 = \frac{1}{1-x(b+1)} - \frac{1}{1-xb}
\end{split}
$$
