<s>[INST] Prove that $(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$ 
For the non-negative real numbers $a, b, c$ prove that $$(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$$

What I did is applying Holder's inequality in LHS:$$(a^2+(\sqrt{2})^2)(b^2+(\sqrt{2})^2)(c^2+(\sqrt{2})^2) \geq (abc + 2\sqrt{2})^2$$ 
Then it suffices to prove that $$(abc+2\sqrt2)^2 \geq 3(a+b+c)^2 \\ \Rightarrow abc+2\sqrt2 \geq \sqrt3(a+b+c)$$ 
But I don't know how to proceed. I also think I applied Holder's Inequality incorrectly.
 [/INST] By C-S $ (a^{2}+2)\left(1+\frac{(b+c)^{2}}{2}\right)\geq(a+b+c)^{2}$.
Hence, it remains to prove that $ (b^{2}+2)(c^{2}+2)\geq3\left(1+\frac{(b+c)^{2}}{2}\right)$,
which is equivalent to $ (b-c)^{2}+2(bc-1)^{2}\geq0$.
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