$x=2\Rightarrow \frac{{{({{2}^{2}}+1)}^{2}}}{{{(2+k)}^{2}}}=\frac{3\times 2+1}{{{(k+2)}^{2}}}+2\Rightarrow \frac{25}{{{(2+k)}^{2}}}=\frac{7+2{{(k+2)}^{2}}}{{{(k+2)}^{2}}}$  ${{(k+2)}^{2}}=t\Rightarrow \frac{25}{t}=\frac{7+2t}{t}\xrightarrow{t\ne 0}7+2t=25\Rightarrow t=9\Rightarrow {{(k+2)}^{2}}=9\Rightarrow \left\{ \begin{align}  & k+2=3\Rightarrow k=1 \\  & k+2=-3\Rightarrow k=-5 \\ \end{align} \right.$