Try rational zeroes first. Factors of 6 are 1,2,3,6 and factors of 22 are 1,11,2,22. Rational zeroes to try are ±1, ±½, ±⅓, ±⅙, ±11, ±11/2, ±11/3, ±11/6, ±2, ±2/11, etc.
f(1)=6-41+93-35-45+22=0 so x-1 is a factor and 1 is a zero.
Synthetic division:
1 | 6 -41 93 -35 -45 | 22
6 6 -35 58 23 | -22
6 -35 58 23 -22 | 0
g(x)=6x4-35x3+58x2+23x-22.
When g(½)=3/8-35/8+29/2+23/2-22=-4+26-22=0, so 2x-1 is another factor, x=½ is a zero.
Synthetic division:
½ | 6 -35 58 23 | -22
6 3 -16 21 | 22
6 -32 42 44 | 0
h(x)=6x3-32x2+42x+44
h(-⅔)=-16/9-128/9-28+44=-16-28+44=0, so 3x+2 is a factor and x=-⅔.
Synthetic division:
-⅔ | 6 -32 42 | 44
6 -4 24 | -44
6 -36 66 | 0
j(x)=6x2-36x+66=6(x2-6x+11). The quadratic has only complex solutions so we have already found the real zeroes: -⅔, ½, 1. The remaining two zeroes are complex:
x2-6x=-11, x2-6x+9=-2, (x-3)2=-2, x=3±i√2.