[tex]x^2+\frac{10}{x^2}+4\sqrt{2}+7=2\left ( 1+\sqrt{2} \right )x+\frac{4\sqrt{2}}{x}\Leftrightarrow \frac{10}{x^2}-\frac{4\sqrt{2}}{x}+\left ( x^2-\left ( 2+2\sqrt{2} \right )x+\left ( 7+4\sqrt{2} \right ) \right )=0\\10-4\sqrt{2}x+\left ( x^2-\left ( 2+2\sqrt{2} \right )x+\left ( 7+4\sqrt{2} \right ) \right )x^2=0[/tex][tex]x^4-\left ( 2+2\sqrt{2} \right )x^3+\left ( 7+4\sqrt{2} \right )x^2-4\sqrt{2}x+10=0\\\left ( x^2+\left ( -1-\sqrt{2} \right )x \right )^2+\left ( 4+2\sqrt{2} \right )\left ( x+\left ( 1-\sqrt{2} \right ) \right )^2+6+2\sqrt{2}=0\\\mathrm{LHS} > 0\Rightarrow x\notin \mathbb{R}[/tex]