The expression ${\left(\sqrt{2{\mathrm{x}}^2+1}+\sqrt{2{\mathrm{x}}^2-1}\right)}^6+{\left(\frac{2}{\sqrt{2{\mathrm{x}}^2+1}+\sqrt{2{\mathrm{x}}^2-1}}\right)}^6$ is a polynomial of degree

We have,

$\begin{array}{c}\frac{2}{\sqrt{2{\mathrm{x}}^2+1}+\sqrt{2{\mathrm{x}}^2-1}}=\frac{2\left(\sqrt{2{\mathrm{x}}^2+1}-\sqrt{2{\mathrm{x}}^2-1}\right)}{\left(2{\mathrm{x}}^2+1\right)-\left(2{\mathrm{x}}^2-1\right)}\\ =\sqrt{2{\mathrm{x}}^2+1}-\sqrt{2{\mathrm{x}}^2-1}\end{array}$

Thus, the given expression can be written as

${\left(\sqrt{2{\mathrm{x}}^2+1}+\sqrt{2{\mathrm{x}}^2-1}\right)}^6+{\left(\sqrt{2{\mathrm{x}}^2+1}-\sqrt{2{\mathrm{x}}^2-1}\right)}^6$

But $(\mathrm{a}+\mathrm{b}{)}^6+(\mathrm{a}-\mathrm{b}{)}^6=2\left[{\mathrm{a}}^6{+}^6{\mathrm{C}}_2{\mathrm{a}}^4{\mathrm{b}}^2{+}^6{\mathrm{C}}_4{\mathrm{a}}^2{\mathrm{b}}^4+{\mathrm{b}}^6\right]$

Therefore, 

$$\begin{gathered} {\left(\sqrt{2{\mathrm{x}}^2+1}+\sqrt{2{\mathrm{x}}^2-1}\right)}^6+{\left(\sqrt{2{\mathrm{x}}^2+1}-\sqrt{2{\mathrm{x}}^2-1}\right)}^6 \\ =2\left[{\left(2{\mathrm{x}}^2+1\right)}^3+15{\left(2{\mathrm{x}}^2+1\right)}^2\left(2{\mathrm{x}}^2-1\right)\left.+15\left(2{\mathrm{x}}^2+1\right)\times {\left(2{\mathrm{x}}^2-1\right)}^2+{\left(2{\mathrm{x}}^2-1\right)}^3\right]\right. \end{gathered}$$

which is a polynomial of degree 6.