First find the rational zeroes. List these: -1, -2, -4, 1, 2, 4. Some of these may be actual zeroes.
We substitute each of these into the polynomial until we get two actual zeroes. It turns out that x=-2 and 2 make f(x)=0. 16+16-20-16+4=0, 16-16-20+16+4=0.
Use synthetic division to divide by -2:
-2 | 1 -2 -5 8 4
1 -2 8 -6 -4
1 -4 3 2 0.
Then divide this result by 2:
2 | 1 -4 3 2
1 2 -4 -2
1 -2 -1 0 = x²-2x-1, x²-2x+1-2=0, (x-1)²=2, x=1±√2.
Zeroes are -2, 2, 1+√2, 1-√2.