The conditions determine q uniquely, since

and therefore b2+b3=3, 2b2+3b3=-3, b2= 12, b3=-9. So .We have q(x)= -1+ x+ b2x2+ b3x3

so q(1)= -1+ 1+ b2+ b3= 3

and q'(x)= 1+ 2b22+ 3b2 so

q'(1)= 1+ 2b2+ 3b2= -2.

You only have two conditions for p(x), so , where s and t are arbitrary real parameters. The solution formulae and can't be correct, since for example this would mean p(0) = t and q(0) = t.Perhaps there is a typo somewhere.