How can I multiply these sums? How do I multiply this? Is it even possible given that one sum is infinite?
$$
x^{n}\left[\,\sum_{r = 0}^{n}\left(-1\right)^{r}x^{mr}\,\right]
\left[\,\sum_{k = 0}^{\infty}{n + k - 1 \choose k} x^{k}\,\right]
$$

You can collect the like powers of $x^k$ like so:
$$
\begin{align}
x^n\left[\sum_{r=0}^n(-1)^rx^{mr}\right]\left[\sum_{k=0}^\infty \binom{n+k-1}{k} x^k\right]
&=\sum_{k=n}^\infty\left[\sum_{r=0}^n(-1)^r\binom{k-mr-1}{k-mr-n}\right]x^k\\
&=\sum_{k=n}^\infty\left[\sum_{r=0}^{\min\left(\left\lfloor\frac{k-n}{m}\right\rfloor,n\right)}(-1)^r\binom{k-mr-1}{n-1}\right]x^k\\
\end{align}
$$
noting that $\displaystyle\binom{n}{k}=0$ when $k\lt0$.
