<s>[INST] 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 $a^2b^2+b^2c^2+c^2a^2$?
 [/INST] $$(a+b+c)=0\implies (a+b+c)^2=0 $$
$$\implies a^2+b^2+c^2+2(ab+bc+ca)=0$$
$$\implies 36+2(ab+bc+ca)=0$$
$$\implies (ab+bc+ca)=-18$$
$$\implies (ab+bc+ca)^2=324$$
just solve this whole square you can find answer
$$\implies a^2b^2+b^2c^2+c^2a^2+2(ab^2c+bc^2a+a^2bc)=324$$
$$\implies a^2b^2+b^2c^2+c^2a^2+2abc(b+c+a)=324$$
since $a+b+c=0$
so
$$\implies a^2b^2+b^2c^2+c^2a^2=324$$
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