<s>[INST] Proving Cauchy inequality involving four expression Show that $$(a^2 + b^2 + c^2) (a^2b^2 +b^2c^2 +c^2a^2) \geq (a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)$$
i should prove this inequality by making it a Cauchy form inequality(as teacher stated).
my problem : this inequality is involving 4 parentheses but in other question i proved previously there was one or two in left side and one in right side.
things i have done so far:
$(a^2 + b^2 + c^2) (a^2b^2 +b^2c^2 +c^2a^2) \geq (a^2b + b^2c + c^2a)^2$ 
 [/INST] Set
$$P = (a^2 + b^2 + c^2) (a^2b^2 +b^2c^2 +c^2a^2) - (a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2).$$
We have
$$P = \frac14\sum c^2\left[a^2+b^2+c^2+(a+b-c)^2\right](a-b)^2 \geqslant 0.$$
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