If this function has rational zeroes they will be factors of 4 which has only 1, 2, 4 as its factors. The first thing to do is to substitute these factors (in positive or negative form) for x in f(x):
f(1)=6, f(-1)=-6, f(2)=0, f(-2)=0, f(4)=84, f(-4)=276. Clearly x=-2 or 2 is a zero.
The factors corresponding to these zeroes are (x+2) and (x-2) so we can use synthetic division to divide by each:
-2 | 1 -2 -5 8 4
1 -2 8 -6 | -4
2 | 1 -4 3 2 | 0 = x3-4x2+3x+2
1 2 -4 | -2
1 -2 -1 | 0 = x2-2x-1.
Finally solve the quadratic:
x2-2x=1, complete the square:
x2-2x+1=2,
(x-1)2=2, x-1=±√2, x=1+√2, 1-√2.
Zeroes are: -2, 2, 1+√2, 1-√2.