If a, b, c are the roots then (x-a)(x-b)(x-c)=x3-5x2-2x+8. The constant term is -abc=8.
The unsigned rational roots are x=1, 2, 4, because 1×2×4=8, the constant term. We can have:
-abc=(-1)(2)(4)=(1)(-2)(4)=(1)(2)(-4)=8. In other words, one root must be negative and the other two positive.
In actual fact, -1 is a true root, because -1-5-2+8=0. Therefore, the other rational roots can only be 2 and 4. Knowing x=-1 is a root we can use synthetic division to reduce the cubic to a quadratic:
-1 | 1 -5 2 8
1 -1 6 | -8
1 -6 8 | 0 =x2-6x+8=(x-4)(x-2), so the other true roots are 2 and 4, which are also the rational roots.