Use the factors x=1 and x=-2 and synthetic division:
x=1:
1 | 2 p q | -14
2 2 p+2 | p+q+2
2 p+2 p+q+2 | p+q-12, so p+q=12 to make the remainder zero.
x=-2:
-2 | 2 p+2 | p+q+2
2 -4 | -2p+ 4
2 p-2 | -p+q+6, so -p+q=-6 to make the remainder zero.
Add the two p-q equations together: 2q=6, q=3 so p=9.
Therefore the third factor is 2x+7.
The original expression evaluates to 2x³+9x²+3x-14.