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 $n=1: 1^3=1^2(2(1)^2-1)$.
2) Assume for $n=k: 1^3+3^3++(2k-1)^3=k^2(2k^2-1)$. Then for $n=k+1$:
$1^3+3^3++(2k-1)^3+(2(k+1)-1)^3=k^2(2k^2-1)+(2(k+1)-1)^3=k^2(2k^2-1)+8k^3+12k^2+6k+1=(k+1)^2(2(k+1)^2-1).$
Thus we proved what we wanted to.