$\displaystyle 6x^{3} - 9x^{2} - 216x + 3$
$\displaystyle 3(2x^{3} - 3x^{2} - 72 + 1)$
$\displaystyle 3(2x^{3} - 3x^{2} - 71)$ ??? Correct? Next move?
$ 6x^3 - 9x^2 -216x +3$
$ 3(2x^3 - 3x^2-72x+1) $
So we actually have: $ 3(2x^3 - 3x^2 - 72x +1 )$
We can use the factor theorem here which states that if f(a)= 0 , (x - a) is a factor; In this case 'a' are all the possible factors of 1 and 1/2. However, f(1)!=0 and f(1/2)!=0 For more information and why a is 1 in this case consult the following link: The Factor Theorem