I've seen something similar around. Since you need the $x$ part to be of the form $(x-a)^n$, you have to extract the $2$ in the $x$:
$(3-2x)^n=(-2)^n \cdot (x-3/2)^n$
Now you have
$c_n = \frac{(-2n^2)^n}{4^n(2n+1)!}$
$c_n = \frac{(-2)^nn^{2n}}{4^n(2n+1)!}$
You now need to evaluate the limit:
$\lim\limits_{n \to \infty} \left| \frac{(-2)^{n+1}{(n+1)}^{2(n+1)}}{(-2)^nn^{2n}}\frac{4^n(2n+1)!}{4^{n+1}(2n+3)!}\right|=$
$\lim\limits_{n \to \infty}\left| \frac{(-2){(n+1)}^{2n}{(n+1)}^2}{n^{2n}}\frac{1}{4(2n+2)(2n+3)}\right|=$
$\frac{1}{2}\lim\limits_{n \to \infty} \left|{\left(\frac{{n+1}}{n}\right)^n}^2\frac{n^2+2n+1}{4n^2+10n+6}\right|=$
$\frac{1}{2}\lim\limits_{n \to \infty}\left| {\left(1+\frac{{1}}{n}\right)^n}^2\frac{n^2+2n+1}{4n^2+10n+6}\right|=$
This limits shouldn't be hard for you, I guess.
$\displaystyle \frac{1}{2}e^2\frac{1}{4}=\frac{e^2}{8}$