How to get value of $a^2b^2+b^2c^2+c^2a^2$ If $a+b+c=0$ and $a^2+b^2+c^2=36$, what is the value of $a^2b^2+b^2c^2+c^2a^2$?

We'll work with what we have until we get what we want:
$$0 = (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2 ac = 36 + 2(ab + bc + ac)$$
And therefore we can conclude that
$$
 - 18 = ab + bc + ac
$$
This in turn means that
$$
324 = (ab + bc + ac) ^2  = a^2b^2 + b^2 c^2 + a^2 c^2 + 2(a^2bc + ab^2 c + abc^2) \\\\
=a^2b^2 + b^2 c^2 + a^2 c^2 + 2abc(a + b + c) = a^2b^2 + b^2 c^2 + a^2 c^2 + 0$$
