First we list the rational zeroes: 1, -1, 2, -2, 3, -3, 6, -6.
If any of these are actual zeroes we only need one to find all the zeroes.
x=1: 1-5+8-6=-2
x=-1: -1-5-8-6=-20
x=2: 8-20+16-6=-2
x=-2: -8-20-16-6=-50
x=3: 27-45+24-6=0, so x=3 is a zero.
Now we use synthetic division to reduce the cubic to a quadratic:
3 | 1 -5 8 -6
1 3 -6
1 -2 2 0 = x²-2x+2.
To solve x²-2x+2=0 we can write this as x²-2x+1+1=0, (x+1)²=-1, x+1=±i.
Therefore x=-1+i and x=-1-i. The zeroes are 3, -1+i, -1-i.