Other Heuristic Method to Prove the Given Expression Greater Than Zero

Given that $x\ne 0$ and $x$ is real that satisfies $\large \sqrt[3]{x^5-20x}=\sqrt[5]{x^3+20x}$. Find the product of all possible values of $x$.

Regarding to the previous blog post(china-imo-contest-problem) that I mentioned how to prove

$a^4+a^3 x+a^2 (x^2+1)+a x^3+x^4+x^2+a x> 0$ in