To solve this equation, we can use the rational root theorem to find possible rational roots. The rational root theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must have a numerator that is a factor of the constant term and a denominator that is a factor of the leading coefficient.

In this case, the constant term is -3 and the leading coefficient is 2, so the possible rational roots are:

±1, ±3, ±1/2, ±3/2

We can try these roots by using synthetic division or long division to see if they are actual roots of the polynomial. After trying all the possible rational roots, we find that x = 1 and x = 3/2 are roots of the polynomial.

Using synthetic division, we can divide the polynomial by (x - 1) and (2x - 3) to get:

2x^3 - 5x^2 + 8x - 3 = (x - 1)(2x^2 - 3x + 3)

(2x^2 - 3x + 3) cannot be factored further, so the solutions to the equation are:

x = 1, x = 3/2, and x = (3 ± i√3)/4 (using the quadratic formula on 2x^2 - 3x + 3)