<s>[INST] If $a+b+c=0$ prove that $ (a^2+b^2+c^2) \times (a^2+b^2+c^2) = 2(a^4+b^4+c^4)$ 
If $a+b+c=0$, for $a,b,c \in\mathbb R$, prove
$$(a^2+b^2+c^2) \times (a^2+b^2+c^2) = 2(a^4+b^4+c^4)$$

What is a good way to do this? 
This question came from answering this slightly harder question. Those answers were somewhat hard to understand for me. To get something easier to digest I made a very similar but easier (lower exponent) question.
 [/INST] Put $c=-a-b$. Then
$$
(a^2+b^2+c^2)^2=4(a^4 + 2a^3b + 3a^2b^2 + 2ab^3 + b^4)=2(a^4+b^4+c^4).
$$
A very similar calculation was given here.
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