How to prove $\sum\limits_{k=0}^n\ (3k^2+2k+1) = n^3 + 5 \begin{pmatrix}n+1\\2\end{pmatrix}+1$ $\sum\limits_{k=0}^n\  (3k^2+2k+1) = n^3 + 5 
\begin{pmatrix}n+1\\2\end{pmatrix}+1$
How would you go on proving this equation? Doesn't have to be induction..

Recall the following identities:
$$\sum_{k=0}^n 1 = n+1$$
$$\sum_{k=0}^n k = \dfrac{n(n+1)}2$$
$$\sum_{k=0}^n k^2 = \dfrac{n(n+1)(2n+1)}6$$
And
$$\sum_{k=0}^n (3k^2+2k+1) = 3 \sum_{k=0}^n k^2 + 2\sum_{k=0}^n k + \sum_{k=0}^n 1$$
Now plug in and obtain what you want.
