Answer
Now, consider the statement: $2$ is a factor of ${{n}^{2}}+3n $
If $ n=1$, the statement is $2$ is a factor of $1+3$
If $ n=2$, the statement is $2$ is a factor of $4+6$
If $ n=3$, the statement is $2$ is a factor of $9+9$
If $ n=k+1$, the statement before the algebra is simplified is $2$ is a factor of ${{\left( k+1 \right)}^{2}}+3\left( k+1 \right)$
If $ n=k+1$, the provided statement after the algebra is simplified is $2$ is a factor of ${{k}^{2}}+5k+4$
Work Step by Step
For the statement $2$ is a factor of ${{n}^{2}}+3n $: if $ n=1$ substituting,
$\begin{align}
& {{n}^{2}}+3n={{1}^{2}}+3\times 1 \\
& =1+3 \\
& =4
\end{align}$
If the value of $ n=2$:
$\begin{align}
& {{n}^{2}}+3n={{2}^{2}}+3\times 2 \\
& =4+6 \\
& =10
\end{align}$
Then, the value $ n=3$ gives
$\begin{align}
& {{n}^{2}}+3n={{3}^{2}}+3\times 3 \\
& =9+9 \\
& =18
\end{align}$
If $ n=k+1$,
$2$ is a factor ${{\left( k+1 \right)}^{2}}+3\left( k+1 \right)$
Further simplification gives,
$\begin{align}
& {{\left( k+1 \right)}^{2}}+3\left( k+1 \right)={{k}^{2}}+2k+1+3k+3 \\
& ={{k}^{2}}+5k+4
\end{align}$
