10.6. Solve in $R$ the equation

$$
\sqrt{2 x^{5}+x^{4}+4 x^{3}+2 x^{2}+2 x+1}+\sqrt{17-2 x+34 x^{2}-4 x^{3}+17 x^{4}-2 x^{5}}=7 x^{2}-8 x+22
$$
Solution: We will apply the inequality $\frac{a+b}{2} \leq \sqrt{\frac{a^{2}+b^{2}}{2}}$, which is true for any real numbers $a$ and $b$.

On $D V A$, the inequality holds:

$$
\frac{\sqrt{2 x^{5}+x^{4}+4 x^{3}+2 x^{2}+2 x+1}+\sqrt{17-2 x+34 x^{2}-4 x^{3}+17 x^{4}-2 x^{5}}}{2} \leq \sqrt{\frac{18 x^{4}+36 x^{2}+18}{2}}=3\left(x^{2}+1\right)
$$

Then

$\sqrt{2 x^{5}+x^{4}+4 x^{3}+2 x^{2}+2 x+1}+\sqrt{17-2 x+34 x^{2}-4 x^{3}+17 x^{4}-2 x^{5}} \leq 6\left(x^{2}+1\right)$.

We will show that on DVA we have $7 x^{2}-8 x+22 \geq 6 x^{2}+6 \Leftrightarrow x^{2}-8 x+16 \geq 0 \Leftrightarrow x \in \mathbb{R}$

Equality holds only for $x=4$.

We verify whether the number 4 is a solution to the equation and convince ourselves that it is a solution.

Answer: $S=\{4\}$