Problem 5. Solve the equation: $2\left(x^{4}+3 x^{2}+6\right)\left(y^{4}-5 y^{2}+12\right)=69$.
Answer: $\left(0 ; \pm \sqrt{\frac{5}{2}}\right)$.

Solution. Note that $x^{4}+3 x^{2}+6 \geqslant 6$ for each $x$, and $y^{4}-5 y^{2}+12=\left(y^{2}-5 / 2\right)^{2}+23 / 4 \geqslant 23 / 4$ for each $y$. Therefore, the left side of the equation is greater than or equal to $2 \cdot 6 \cdot 23 / 4=69$, and equality is achieved only when $x=0$ and $y^{2}=5 / 2$. This gives the answer.