<s>[INST] Prove that if $a,b,$ and $c$ are positive real numbers then $(a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)}.$ 
Prove that if $a,b,$ and $c$ are positive real numbers then $$(a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)}.$$

This looks like a simple question. We can apply AM-GM twice to get $(a+b)(a+c) \geq 4a\sqrt{bc}$. Then how do I use that fact to get $(a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)}$?
 [/INST] We have
$$(a+b)(a+c) = a(a+b+c) +bc.$$
We can now apply AM-GM.
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