I am getting -14 and -4.

has 2 real solutions and two non-real. Are you sure you typed it correctly. One sign is wrong then it's all wrong.

If you factor

As you can see, two soluitons are 3/2 and -1/2

Plug these into the quartic and you get:

Solve these two for p and q and you get q=-14 and p=-4

Sub those in and you get:

The factorization you said was in the book when expanded gives: