We need values of x that would make the given equal to zero. Once we know these values, then we use the Factor Theorem to get the corresponding factors. If, for example, we substitute 5 for x and the given becomes equal to zero, then (x-5) is a factor. If -3 does that, then (x+3) is a factor.

To limit the choices for the values of x that we will try, we can make use of the Theorem on Possible Rational Roots. This theorem states that the possible rational roots include factors of the constant (in your given problem, this is the number -8). Therefore, positive and negative 1, 2, 4 and 8 are possible rational roots.

Let’s substitute some of these values.

If we try x = 1, we get:

(1)^3 + 5(1)^2 + 2(1) – 8

= 1 + 5 + 2 -8 = 0

Therefore 1 is a zero, and (x-1) is a factor.

Let’s try x = -1:

(-1)^3 + 5(-1)^2 + 2(-1) – 8

= -1 + 5 – 2 – 8 = -6

Therefore -1 is NOT a zero and (x+1) is NOT a factor.

If you continue to do this for the other values, you’ll get -2 and -4 are zeroes, which means that (x+2) and (x+4) are factors.

This is actually the short version of the longer and more comprehensive explanation you might need, but I hope this helps.