induction - Inductive proof of the degree of a polynomial

Here is the problem:

Assume that there is a polynomial $P(x)$ of degree 4 such that for all $N \in \mathbb{N}$,

$$P(N) = \sum\limits_{n=0}^N n^3$$

Find the polynomial. Use induction to prove that the formula is correct.

...............

Not sure where to start on this, but for the base case, I did $n=0$ results in $0^3=0$. How can I prove this has degree $4$, since $0^5=0$, for example? Also, how can I prove it for N in the inductive step?

Also... before I even get there, I'm puzzled about the polynomial. I know it's something like:

$$ax^4+bx^3+cx^2+dx+e = x^3 + (x-1)^3 + (x-2)^3 + \cdots + 1$$

and you get an $x^4$ term on the RHS because you have $x$ number of times $x^3$, but I don't know where to go from there to find the polynomial.

Thank you!