Factorize the polynomial \(2x^3 - 5x^2 - 23x + 30\).

Step 1 ：Step 1: We start by trying to factor by grouping. However, this polynomial doesn't seem to be factorable in that way. So, we will need to use another method. Let's try factoring by synthetic division. We know that the possible rational roots of the polynomial are the factors of the constant term, 30, divided by the factors of the leading coefficient, 2. So, the possible rational roots are \(\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30, \pm 1/2, \pm 3/2, \pm 5/2, \pm 15/2\).

Step 2 ：Step 2: We test these possible roots by synthetic division. We find that -2, -3/2, and 5 are roots of the polynomial.

Step 3 ：Step 3: Thus we can write the polynomial as \((2x + 4)(2x + 3)(x - 5)\).