3. Solve the equation

$$
2 x+2+x \sqrt{x^{2}+1}+(x+2) \sqrt{x^{2}+4 x+5}=0 .
$$
Answer: -1.

Solution. Let $f(x)=x\left(1+\sqrt{x^{2}+1}\right)$. The original equation can be rewritten as $f(x)+f(x+2)=0$. Note that the function $f(x)$ is odd. It is increasing on the positive half-axis (as the product of positive increasing functions). Due to its oddness, it is increasing on the entire real line. Further, we have

$$
f(x)=-f(x+2) ; \quad f(x)=f(-x-2)
$$

Since an increasing function takes each of its values exactly once, it follows that $x=-x-2$, from which $x=-1$.

Remark. The monotonicity of the left side of the original equation could also have been established using the derivative.

Evaluation. 13 points for a correct solution. If the answer is guessed but not proven that there are no other solutions, 3 points.