Using a Known Factor

The factor theorem finds one factor; long division finds the rest. Take p(x) = x^3 - 2x^2 - 5x + 6. Try small numbers: p(1) is 1 - 2 - 5 + 6, which is zero, so x - 1 is a factor.

Divide it out: (x^3 - 2x^2 - 5x + 6)/(x - 1) is x^2 - x - 6, with nothing left over. What remains is a quadratic, which you already know how to factor: x^2 - x - 6 is (x - 3)(x + 2). So

x^3 - 2x^2 - 5x + 6 = (x - 1)(x - 3)(x + 2)

Which numbers should you try? When p starts with x^3 and has whole-number coefficients, any whole-number root divides the constant term, because every other term has a factor of x. Here the constant is 6, so try 1, 2, 3 and 6 and their negatives. One success is enough: the division turns the cubic into a quadratic.