I don't understand generating functions very well and I was told in one of my questions that the coefficient of $t^N$ in the product between $$\frac{1}{(1-t)^2}\,\exp\left(-\frac{tx}{1-t}\right)$$ and $$\frac{2-4t+2t^2-t^2x^2-t^2 x^3}{2t^2 x^2}+\frac{(1-t)(tx+t-1)}{t^2 x^2}\,\exp\left(\frac{tx}{1-t}\right)$$ can be computed through $$\sum_{n\geq 0}L_n(x)\,t^n = \frac{1}{1-t}\,\exp\left(-\frac{tx}{1-t}\right)$$ where $L_n(x)$ is the Laguerre polynomial. How is this done?
I would be perfectly satisfied with an explanation on how to solve it rather than the solution itself.
The product is
\begin{align} &\frac{2-4t+2t^2-t^2x^2-t^2 x^3}{2t^2 x^2(1-t)^2}\exp\left(-\frac{tx}{1-t}\right)+\frac{tx+t-1}{t^2 x^2(1-t)}\\ ={}& \frac{2-4t+2t^2-t^2x^2-t^2 x^3}{2t^2 x^2(1-t)}\sum_{n\ge0}L_n(x)t^n+\frac{tx+t-1}{t^2 x^2(1-t)} \\ ={}& \frac1{2t^2x^2(1-t)}\left( \left(2-4t+2t^2-t^2x^2-t^2 x^3\right)\sum_{n\ge0}L_n(x)t^n+2(tx+t-1)\right) \;. \end{align}
Except for the factor $1-t$ in the denominator, the coefficient of $t^N$ is straightforward to extract. Multiplying by $1/(1-t)=\sum_kt^k$ sums the coefficients, so the coefficient of $t^N$ is the sum of the coefficients up to and including $t^N$ in the rest of the expression.