Answer
See below.
Work Step by Step
Proofs using mathematical induction consists 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:
1) For $n=1: 3=(1)(1+2)$.
2) Assume for $n=k: 3+4+...+(2k+1)=k(k+2)$. Then for $n=k+1$:
$3+4+...+(2k+1)+(2k+3)=k(k+2)+(2k+3)=k^2+2k+2k+3=(k+1)(k+3)=(k+1)((k+1)+2).$
Thus we proved what we wanted to.