To solve the equation x^3 - 3x + 2 = 0, you can use the Rational Root Theorem or synthetic division to find the possible rational roots.

The possible rational roots are the factors of the constant term (2) divided by the factors of the leading coefficient (1).

The factors of 2 are ±1 and ±2, and the factors of 1 are ±1.

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

By trying these values, we find that x = -1 is a root of the equation.

Using synthetic division, we can divide the equation by (x + 1) to find the other roots.

Performing the synthetic division, we get:

-1 | 1 -3 0 2
-1 4 -4
________________
1 1 4 -2

The result is x^2 + x + 4 - 2/(x + 1) = 0.

Now, we have a quadratic equation x^2 + x + 4 - 2/(x + 1) = 0.

To solve this equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = 1, b = 1, and c = 4.

Plugging these values into the quadratic formula, we get:

x = (-1 ± √(1^2 - 4(1)(4)))/(2(1))
= (-1 ± √(1 - 16))/2
= (-1 ± √(-15))/2

Since the discriminant (b^2 - 4ac) is negative, the roots are complex numbers.

Therefore, the solutions to the equation x^3 - 3x + 2 = 0 are x = -1 and x = (-1 ± √(-15))/2.