$\displaystyle 4x^{3}-6x^{2}-72x$
$\displaystyle 2x(2x^{2} - 3x - 36)$
?? hint on next move
next move to do what? Are you trying to factor it completely?
It can't be factored further over the rationals.
If you want to find the all the factors over the real numbers apply the quadratic formula to find the 2 roots, $r_1$ and $r_2$, of $2x^2-3x-36$
and your final factorization will be
$2x(x-r_1)(x-r_2)$
Agree with romsek, the expression cannot be factored further. You have to use the quadratic formula to find the roots of the expression within the brackets. Also, the roots will not be rational numbers since the discriminant is not a perfect square.