Prove by mathematical induction that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } \left( 5 r ^ { 4 } + r ^ { 2 } \right) = \frac { 1 } { 2 } n ^ { 2 } ( n + 1 ) ^ { 2 } ( 2 n + 1 )$$
Use the result given in part (a) together with the List of formulae (MF19) to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { r } ^ { 4 }\) in terms of \(n\), fully factorising your answer.