5. Solve the system of equations $\left\{\begin{array}{l}x^{4}+\frac{7}{2} x^{2} y+2 y^{3}=0 \\ 4 x^{2}+7 x y+2 y^{3}=0\end{array}\right.$.
Solution. Consider the function $f(t)=t^{2}+\frac{7}{2} t y+2 y^{3}$. Under the condition that the equalities of the original system are satisfied, its roots will be $t_{1}=x^{2}$ and $t_{2}=2 x$. If $t_{1}=t_{2}$, then $x_{1}=0, x_{2}=2$. From this, we find $y_{1}=0, y_{2}=-1$. If $t_{1} \neq t_{2}$, then using Vieta's theorem, we get

$t_{1} \cdot t_{2}=2 y^{3} \Leftrightarrow 2 x^{3}=2 y^{3} \Leftrightarrow x=y$. Substituting into the original system, we find the third solution $\left(-\frac{11}{2} ;-\frac{11}{2}\right)$

Answer: $(0 ; 0),(2 ;-1),\left(-\frac{11}{2} ;-\frac{11}{2}\right)$