4. Solve the inequality $4+x^{2}+2 x \sqrt{2-x^{2}}<8 \sqrt{2-x^{2}}+5 x$. (20 points)
Solution. Rewrite it as $4+x^{2}-5 x+2 x \sqrt{2-x^{2}}-8 \sqrt{2-x^{2}}0\end{array}\right.$ Solve the second inequality of the system

$(x-1)+2 \sqrt{2-x^{2}}>0 \Rightarrow 2 \sqrt{2-x^{2}}>1-x \Rightarrow\left[\begin{array}{l}\left\{\begin{array}{l}1-x1-2 x+x^{2}\end{array}\right.\end{array}\right.$ $\left[\begin{array}{l}\sqrt{2} \geq x \geq 1, \\ 5 x^{2}-2 x-7<0,\end{array} \Rightarrow\left[\begin{array}{l}\sqrt{2} \geq x \geq 1, \\ \left\{\begin{array}{l}x \leq 1, \\ -1<x<7 / 5,\end{array}\right.\end{array} \Rightarrow x \in(-1 ; \sqrt{2}]\right.\right.$

Answer $(-1 ; \sqrt{2}]$