p=q+2 making 2p+3q=-26 become:
2(q+2)+3q=-26 perform indicated multiplication on left side
2q+4+3q=-26 combine like terms on left side
5q+4=-26 subtract 4 from both sides
5q=-30 divide both sides by 5
q=-6, and from earlier we know that p=q+2 so
p=-4
Thus (p,q)=(-4,-6)