i was wondering if someone could check my work...

i am supposed to rewrite the following series as a single power series...

$\displaystyle 4\sum_{n=2}^{\infty }n(n-1)c_{n}x^{n-2}-3\sum_{n=1}^{\infty}nc_{n}x^{n-1}+\sum_{n=0}^{\infty}c_{n}x^{n}$

here is what i got as an answer...did i do it right?

$\displaystyle 8c_{2}-3c_{1}+c_{0}+4\sum_{k=1}^{\infty}(k+2)(k+1)c_{k+2} x^{k}-3\sum_{k=1}^{\infty}(k+1)c_{k+1}x^{k}+\sum_{k=1}^{ \infty}c_{k}x^{k}$