Answer
See below.
Work Step by Step
Proofs using mathematical induction consist of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number. Then we prove that then the statement also holds for $n + 1$.
Hence, here we have:
1) For $n=1: 2^3=2(1^2)((1)+1)^2$.
2) Assume for $n=k: 2^3+4^3++(2k)^3=2k^2(k+1)^2$. Then for $n=k+1$:
$ 2^3+4^3++(2k)^3+(2(k+1))^3=2k^2(k+1)^2+(2(k+1))^3=2k^2(k+1)^2+8k^3+24k^2+24k+8=2(k+1)^2(k+2)^2.$
Thus we proved what we wanted to.