3B. If $a+b+c=0$ and $a^{2}+b^{2}+c^{2}=6$, calculate the value of the expression

$$
a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}
$$
Solution. From the fact that

$$
(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2(a b+b c+c a)
$$

and from the condition of the problem, we have that $a b+b c+c a=-3$. If we use the formula for the square of a trinomial once again, we have

$$
(a b+b c+c a)^{2}=a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}+2 a b c(a+b+c)=9
$$

From this,

$$
a^{2} b^{2}+b^{2} c^{2}+c^{2} a^{2}=9
$$

## 2nd year