Answer
See below.
Work Step by Step
By substituting in:
for $n=1$ the statement says that $2$ is a factor of $1^2+3\cdot1=1+3=4$
for $n=2$ the statement says that $2$ is a factor of $2^2+3\cdot2=4+6=10$
for $n=3$ the statement says that $2$ is a factor of $3^2+3\cdot3=9+9=18$
for $n=k+1$ the statement says that $2$ is a factor of $(k+1)^2+3\cdot(k+1)$, which can be simplified to $k^2+2k+1+3k+3=k^2+5k+4$
