<s>[INST] Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$ 
Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $$\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$$

I have attempted multiple times in this question and the only method that I can think of is by adding up the fractions and expanding, but took too long and I eventually got stuck. Is there any other method I can use to help me in this question?
 [/INST] Hint $${ab+4\over a+2} = {2ab+8\over 2a+4} = {2ab+abc\over 2a+4} = {ab\over 2}\cdot{2+c\over a+2}$$
Similary for the rest of terms. Now use AM-GM for 3 terms...
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