Assume you mean f(x)=x⁴-x³-6x²+4x+8.
Define set P as all the factors of the constant so P={ 1, 2, 4, 8 }.
Define set Q as all the factors of the coefficient of x⁴ so Q={ 1 }.
Create all possible ±p/q where p is in P and q is in Q:
1, 2, 4, 8, -1, -2, -4, -8.
Use synthetic division to decide which, if any, are actual factors:
-2 | 1 -1 -6 4 8
1 -2 6 0 -8
1 -3 0 4
-1 | 1 -1 -6 4 8
1 -1 2 4 -8
1 -2 -4 8
2 | 1 -1 -6 4 8
1 2 2 -8 -8
1 1 -4 -4
Only those divisions have been shown which confirm an actual zero (to save space and time in the answer). The rational zeroes are -2, -1, 2 (duplicate). (x-2)² is a factor because:
2 | 1 1 -4 -4
1 2 6 4
1 3 2 2 repeats as a zero.