I got as far as (x+2)(x^2-2x-2). I see that the answer in the back of my book is -2, 1 +/- sq(3). (the sq(3) meaning the square root of 3). I get how I get the -2, but how do we get the answer of 1 +/- sq(3) from (x^2-2x-2)?
The rational root theorem gives all the possible rational roots (integer divided by an integer). The possible rational roots are all factors of p (-4, the constant) divided by all factors of q (1, the coef on the high order term) both positive and negative. Further, Descartes' rule of signs says there are either 0 or 2 positive rational roots. So, the possible positive rational roots are 1, 2, and 4. None of those work, so there are no positive RATIONAL roots. If we look at -1, -2, and -4; only -2 works, so it is the only rational root. If you divide the equation by (x+2, which is x minus the root), that leaves you with a quadratic equation, which can be solved by the quadratic formula. This results in the other 2 non-rational roots (1+sqrt(3), 1-sqrt(3)).