To solve the equation x^3 - 3(x + 2) + 8 = 0, we can start by simplifying the expression:

x^3 - 3(x + 2) + 8 = 0
x^3 - 3x - 6 + 8 = 0
x^3 - 3x + 2 = 0

Now, we can try to factor the equation. However, this equation does not have any obvious factors. Therefore, we can use the rational root theorem to find possible rational roots.

The rational root theorem states that if a polynomial equation has a rational root p/q, where p is a factor of the constant term (2 in this case) and q is a factor of the leading coefficient (1 in this case), then p/q is a possible rational root.

The factors of 2 are ±1 and ±2, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1 and ±2.

We can test these possible roots by substituting them into the equation:

For x = 1:
1^3 - 3(1) + 2 = 0
1 - 3 + 2 = 0
0 = 0

For x = -1:
(-1)^3 - 3(-1) + 2 = 0
-1 + 3 + 2 = 0
4 = 0

For x = 2:
2^3 - 3(2) + 2 = 0
8 - 6 + 2 = 0
4 = 0

For x = -2:
(-2)^3 - 3(-2) + 2 = 0
-8 + 6 + 2 = 0
0 = 0

From these tests, we can see that x = 1 and x = -2 are the roots of the equation.

Therefore, the solutions to the equation x^3 - 3(x + 2) + 8 = 0 are x = 1 and x = -2.