Let $m$ be a fixed rational number. Are there any rational solutions to this equation

$$x^{8}-m^{2}x - m = 0$$ ?

My attempt: The equation can be rewritten as $$x^{9} = mx(mx+1)$$

Hence we should have $mx = a^{9}$ and $mx+1 = b^{9}$ for some rational $ab$, so that $b^9 - a^9 = 1$. By Fermat's Last Theorem, this yields $ab=0$ as the only rational solutions.

However, i'm not sure if $mx$ and $mx+1$ should necessarily be of the form that i stated.