(a) Prove by mathematical induction that, for all positive integers \(n\),

\(\sum_{r=1}^{n} (5r^4 + r^2) = \frac{1}{2} n^2 (n+1)^2 (2n+1).\)

(b) Use the result given in part (a) together with the List of formulae (MF19) to find \(\sum_{r=1}^{n} r^4\) in terms of \(n\), fully factorising your answer.