The equation x

^{5} –23x

^{3} –66x

^{2} +px +q =0 has three non-zero roots, and two of them are integers, with multiplicity 2.

Find p and q and determine all the roots of the given equation.

(In reply to

Who needs p and q? (SPOILER) by Old Original Oskar!)

There is a sign problem in the "Solving..." paragraph: for b=+/-1, then a=-2, or for b=+/-2, then a=-1, so a=-2 and b=-1, and the roots come to be -2 (twice), -1 (twice), and 6 (once).