x^4 + 2x^3 - 3x^2 - 28x - 24 = 0
The factors of 24 are: 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 24, -24
The factors of 1 are: 1, -1
Note that 24 is the constant, and 1 is the coefficient of x^4
By the rational root theorem, if a root is rational, it will have to be of the form:
(factor of 24) / (factor of 1).
Thus, the possible rational roots are: 1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 8, -8, 12, -12, 24, -24
You can try substituting the possible roots in till you get a correct root.
The correct roots will be x = -1 and x = 3 The factors will thus be (x + 1) and (x - 3).
Using long division on x^4 + 2x^3 - 3x^2 - 28x - 24 with (x + 1), and then with (x - 3) a second time, you will have x^2 + 4x + 8, which has no real roots.
Hence, x = -1 and x = 3 are the only real roots.