<s>[INST] Show that $(a+2b+\frac{2}{a+1})(b+2a+\frac{2}{b+1})\geqslant16$ if $ ab\geqslant1$ $a$, $b$ are positive real numbers.
Show that $$\left(a+2b+\frac{2}{a+1}\right)\left(b+2a+\frac{2}{b+1}\right)\geqslant16$$
if
$$ ab\geqslant1$$ 
Should I use AM-GM inequality? If yes, where?
 [/INST] Expand to get:
$$ab + 2a^2 + \frac{2a}{b+1} + 2b^2 + 4ab + \frac{4b}{b+1} + \frac{2b}{a+1} + \frac{4a}{a+1} + \frac{4}{(a+1)(b+1)} \ge 9 + \frac{2a^2 + 2a + 4ab + 4b + 2b^2 + 2b + 4ab + 4a + 4}{(a+1)(b+1)} \ge 9 + 7 = 16$$
The last inequality follows as:
$$2a^2 + 2a + 4ab + 4b + 2b^2 + 2b + 4ab + 4a + 4 \ge 7(a+1)(b+1)$$
$$2a^2 + 2b^2 + ab \ge a+ b + 3$$
Now it's enough to prove that $2a^2 + 2b^2 \ge a + b + 2$. From the condition we have by AM-GM $a+b \ge 2\sqrt{ab} \ge 2$. Hence by using the quadratic mean inequality:
$$2a^2 + 2b^2 \ge (a+b)^2 \ge 2(a+b) \ge a+b+2$$
Hence the proof.
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