Try the rational zeroes first. We only need to solve the equation.
The factors of the coefficient of the highest degree of x are 1 and 2.
The factors of the constant are 1, 2, 3, 6.
The rational zeroes are positive and negative values of :
1, 2, 3, 6, ½, 1½.
So let's try 1 and -1: 2-9+6+11-6=4; 2+9+6-11-6=0⇒x=-1 is a zero.
Divide by this zero (synthetic division):
-1 | 2 -9 6 11 -6
2 -2 11 -17 | 6
2 -11 17 -6 | 0 = 2x3-11x2+17x-6.
Now we can continue to look for another rational zero from the list. Try x=2:
16-44+34-6=0, so x=2 is another zero. Divide by this zero:
2 | 2 -11 17 -6
2 4 -14 | 6
2 -7 3 | 0 = 2x2-7x+3=(2x-1)(x-3), giving us the two remaining zeroes: ½ and 3.
SOLUTION: Zeroes are -1, ½, 2, 3.