Graph the cubic equation. The places where the graphed curve crosses the X axis indicates a real root. If there are complex roots, they always come in pairs, so if the cubic equation has complex roots, there will be two complex roots and one real root. If there is a double root, such as X^3 + X^2 - X - 1 = (X - 1)(X + 1)^2, the graphed curve will touch the X axis at one point.
Use the first and last numbers in the cubic equation to generate candidate factors. The factors have the same roots as the cubic and are much easier to solve. The first and last numbers of the factors will be factors of the first and last numbers in the cubic. For example, the first number in X^3 - 7X - 6 is 1 -- the coefficient of X^3 -- which only has one factor: 1. The last number is 6 which has factors 1, 2, 3 and 6. The candidate factors are X - 1, X + 1, X - 2, X + 2, X - 3, X + 3, X - 6 and X + 6.
Try each of the candidate factors to see which of the factors divide the cubic without leaving a remainder. For the cubic X^3 - 7X - 6 we find that X^3 - 7X - 6 = (X + 1)(X + 2)(X - 3). The roots of the cubic are the same as the roots of the factors -- the solutions of the equations X + 1 = 0, X + 2 = 0 and X - 3 = 0. The roots are -1, -2 and 3.