Induction proof of divisibility I am trying to solve a proof by induction, which says

For all $n \in \mathbb{N}_0$, 
  $$ ((2n+1)^2 - 2^{n+1}(2n+1) + 2^{2n+1}) \mid ((2n+1)^4 + 4^{2n+1}). $$

Every try ended in a dead end. Does anybody have a hint?

Notice that we can expand the right hand side as
\begin{equation}
\begin{split}
(2n+1)^4+4^{2n+1} &= \left((2n+1)^2+2^{2n+1}\right)^2-2^{2n+2}\cdot(2n+1)^2 \\&= \left((2n+1)^2-2^{n+1}\cdot(2n+1)+2^{2n+1}\right)\left((2n+1)^2+2^{n+1}\cdot(2n+1)+2^{2n+1}\right)
\end{split}
\end{equation}
Hence $$\left((2n+1)^2-2^{n+1}\cdot(2n+1)+2^{2n+1}\right)| \left((2n+1)^4+4^{2n+1}\right) $$
as desired.
