Answer
See the explanation below.
Work Step by Step
$\begin{align}
& {{S}_{1}}:2=2{{\left( 1 \right)}^{2}} \\
& 2=2\ \text{is true}\text{.} \\
& {{S}_{k}}:2+6+10+....+\left( 4k-2 \right)=2{{k}^{2}} \\
& {{S}_{k+1}}:2+6+10+....+\left( 4k-2 \right)+\left( 4k+2 \right)=2{{\left( k+1 \right)}^{2}} \\
\end{align}$
And add $\left( 4k+2 \right)$ to both sides of ${{S}_{k}}$:
$2+6+10+....+\left( 4k-2 \right)+\left( 4k+2 \right)=2{{k}^{2}}+\left( 4k+2 \right)$
Then, simplify the right-hand side:
$\begin{align}
& 2{{k}^{2}}+4k+2=2({{k}^{2}}+2k+1) \\
& 2{{k}^{2}}+4k+2=2{{\left( k+1 \right)}^{2}} \\
& \text{If }{{S}_{k}}\ \text{is true, then }{{S}_{k+1}}\ \text{is true}\text{. } \\
\end{align}$
