Mathematical Induction problem

problem #14 on here:

https://lh4.googleusercontent.com/_2...406_022154.jpg

This is what ive done so far:

1.) prove that the 2 equations are equal by testing P(n) at 0. They are, so i proceed to the inductive step and rewrite the equation in terms of k.

2.) I then try nd show for all integers > or equal to k if P9K) is true then P(k+1) is also true.

I do this by equating $\displaystyle \left( k+1 \right)2^{\left( k+1 \right)+2}+2$

with $\displaystyle \sum_{i=0}^{k+1}{k2^{k}}+k^{k+1}$

then attempt to turn it into what got for P(k+1), but im pretty sure i messed up making the last term explicit.