# 3. Solve the system of equations

$$
\left\{\begin{array}{l}
x^{5}+3 x^{4}+5 x^{3}+5 x^{2}+6 x+2=0 \\
x^{3}+3 x^{2}+4 x+1=0
\end{array}\right.
$$
Solution. Transform the first equation:

$$
\begin{aligned}
& \left(x^{5}+3 x^{4}+4 x^{3}+x^{2}\right)+x^{3}+4 x^{2}+6 x+2=0 \Rightarrow \\
& x^{2}\left(x^{3}+3 x^{2}+4 x+1\right)+\left(x^{3}+3 x^{2}+4 x+1\right)+x^{2}+2 x+1=0
\end{aligned}
$$

Considering that $x^{3}+3 x^{2}+4 x+1=0$, we get $x^{2}+2 x+1=0$. Solving the last equation, we find $x=-1$. Therefore, this value of $x$ could be a solution to the system. Upon verification, we confirm that $x=-1$ does not satisfy the system of equations.

Answer: the system has no solutions.