Factorization is an art (8)

Let $\large \sqrt[7]{x^{5} + 20x^{3}} = \sqrt[5]{x^{7} - 20x^{3}} = a$

Then, we have

$\large x^{5} + 20x^{3} = a^{7}$ and $\large x^{7} - 20x^{3} = a^{5}$

$\large \Rightarrow x^{5} + x^{7} = a^{5} + a^{7}$

$\large \Rightarrow x = a$ (since the function $f(x) = x^{5} + x^{7}$ is monotonically increasing for all x)

$\large \Rightarrow x^{5} + 20x^{3} = x^{7}$

$\large \Rightarrow x^{4} - x^{2} - 20 = 0$ (since $x \neq 0$ )

$\large \Rightarrow (x^{2} - 5)(x^{2} + 4) = 0$

$\large \Rightarrow x = \pm \sqrt{5}$

$\Rightarrow$ the product of all possible values of x is -5.