Solving AM-GM without expanding

Question

Prove that $(p^2+p+1)(q^2+q+1)(r^2+r+1)(s^2+s+1) ≥ 81pqrs$ for all $p, q, r, s ≥ 0$. I am not sure how to solve this problem. I know that this is supposed to be an AM-GM question, so I tried expanding it. It didn't take long for me to realize that expanding is probably not the way to go to solve this problem. I also tried taking the fourth root of both sides, but that didn't solve anything either. How can I prove this? Any help would be appreciated.

Answer 1

I have the original problem in my math book, "The Art of Problem Solving: Intermediate Algebra", and it says that all $p,q,r,s$ are positive, which is important to note to solve this problem.

Apply AM-GM to the terms $p^2,p,1$. We get $\frac{p^2+p+1}3 \ge \sqrt[3]{p^2*p*1}$. Simplifying and rearranging we get $p^2+p+1 \ge 3p$. Now apply the same logic to each of the other expressions to get $q^2+q+1 \ge 3q$, $r^2+r+1 \ge 3r$, and $s^2+s+1 \ge 3s$. Since all of $p,q,r,s$ are positive, then all of these expressions are positive, too, so we can safely multiply all the inequalities together to get $(p^2+p+1)(q^2+q+1)(r^2+r+1)(s^2+s+1) \ge 81pqrs$ as desired.