How do I solve $x^3= (3mx-4m^3)^2 $?

Question

How do I solve $x^3= (3mx-4m^3)^2 $? This is actually a step in the answer of a coordinate geometry problem I'm solving. The answers are $4m^2$ and $m^2$. Is there a quick and easy way to figure it out?

Answer 1

Because of the rhs, let $x=m^2y$ (assuming $m\neq 0$) to make the equation $$m^6y^3=m^6 (4-3 y)^2 \implies y^3-(4-3y)^2=0$$ Expand $$y^3-9 y^2+24 y-16=0$$ By inspection, $y=1$ is a root. Long division reduces to $y^2-8 y+16=(y-4)^2=0$