Let x=m3, and we have x3-x-2 as the trinomial.
There are various ways of solving this (which doesn't factorise rationally).
One way is to use Newton's Method. Let f(x)=x3-x-2, then the derivative f'(x)=3x2-1.
f(1)=-1 and f(2)=4, so a root (a zero) must lie between x=1 and 2.
Newton's Method is iterative: xn+1=xn-f(xn)/f'(xn).
We start with an approximate root, so let x0=1.
x1=2, x1=18/11, x2=1.530..., x3=1.521..., x4=..., xn=1.5213797 approx after a few more iterations.
m=∛x=1.150127 approx. So m-1.150127 is a factor. The other factor is a quadratic with irrational coefficients. There are no more real solutions.