Generating function into polynomail

Question

i) Find a generating function expression of a sequence with terms $$d_n=\sum_{p=0}^n p^3$$ using operations on the geometric series $\sum_{n\geq 0} x^n$

ii) Derive a polynomial (in $n$) expression for $d_n$.

for i) I got $x(1+4x+x^2)/(1-x)^5$ but I'm confused what to do for ii), how does one derive that?

Answer 1

Hint:

$$\frac{1}{(1-x)^5} = (1-x)^{-5}=\sum_{n=0}^{\infty} \binom{n+4}{4} x^n$$ by the Binomial Theorem.