<s>[INST] finding the coefficient of $x^{14}$ in the expression: $\frac{5x^2-x^4}{(1-x)^3}$ I have a homework question which requires me to find the coefficient of $x^{14}$ in the expression: $\dfrac{5x^2-x^4}{(1-x)^3}$
I have not figured out a way to do this (I believe this is because my algebra is weak).
This is a question in a combinatorics class and has to do with Generating Functions if that is any help.
Help is greatly appreciated. Thanks, Jason
 [/INST] First pull a $5$ and an $x^2$ from the rational function. This gives you
$$
\frac{5x^2\left(1 - \left(\frac{1}{5}x^2\right) \right)}{(1-x)^3}.
$$
Then, use the expansions
$$
(1+x^m)^n = 1 - \binom{n}{1}x + \binom{n}{2}x^2 + \dots + (-1)^k\binom{n}{k}x^{km} + \dots + (-1)^n \binom{n}{n}x^{nm}
$$
and 
$$
\frac{1}{(1-x)^n} = 1  + \binom{1 + n -1}{1}x  + \binom{2 + n -1}{2}x^2 + \dots +  \binom{r + n -1}{r}x^r + \dots.
$$
To find the coefficient of $x^{12}$ (Why $x^{12}$?)
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