Proving the an expression is larger than a simplified quadratic

Question

Let p and q be positive real numbers. Prove that $$ (p + 2)(q+2)(p+q) \ge 16pq $$

Any explanation/answer would be extremely helpful. Thanks : )

Answer 1

Apply AM-GM inequality to each factor: $$p+2\ge2\sqrt{2p}$$ $$q+2\ge2\sqrt{2q}$$ $$p+q\ge2\sqrt{pq}$$

Multiply the three inequalities and you are done.

Answer 2

Holder's inequality is applicable as well.

$$(2+p)(q+2)(p+q) \ge \left(\sqrt[3]{2qp}+\sqrt[3]{p2q}\right)^3=16pq$$

with equality iff $\frac{2}{q}=\frac{p}{2}$ and $\frac{q}{p}=\frac{2}{q}$ and $\frac{2}{p}=\frac{p}{q}$.

I.e. $pq=4,\,q^2=2p,\,p^2=2q$, i.e. $p=q=2$.