reference request - General technique to find partial sum formula for series such as $sum n^3/2^n$

Background. Working in a secondary school class on random walks that could only head in two directions (e.g., South and West) we stumbled upon the following summation to be evaluated:

$$\sum_{n=0}^{\infty} \frac{(n-1)n(n+1)}{2^n}$$

We were able to "solve" this in three ways: using some probabilistic intuition, plugging the formula directly into Mathematica/Wolfram Alpha, and using the identity:

$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots$$

differentiated a few times over to find (verification pasted from Wolfram Alpha):

after which plugging in $x = 1/2$ gives roughly the series about which we were curious.

Another instructor was able to tinker with the above-mentioned series and figure out a closed form for the $m$th partial sums, which could then be verified by induction (after which taking the limit as $m \rightarrow \infty$ resolves the matter). In fact, Wolfram Alpha produces the closed form for this partial sum immediately upon input. For example, inputting the series above yields:

Question: Given an infinite series that consists of the ratio of a polynomial in $n$ (numerator) to a constant raised to some power that is linear in $n$ (denominator) what is a general technique to produce the closed formula corresponding to the series' $m$th partial sums?

Given the context, material at the level of strong secondary mathematics (or early undergraduate mathematics) would be ideal, but mathematics at any level - references, related problems and/or solutions, and extensions - are all welcome, too.

I would especially appreciate answers that "fill in" all details, e.g., by including a worked example, so as to make this post more pedagogically effective in a self-contained manner.