You're on the right track. Your derivative should be b+2ct, not t+2ct. Evaluating q' and q at 1 and 2 respectively, you get b+2c = a+2b+4c. This is a linear system of three unknowns and one equation, so it's just basic linear algebra from here.
Find a basis of the linear space and thus determine the dimension.
Where q is a subset of Q_2: all polynomials of degree less than or equal to 2. I had to change it from p to q so the forum didn't make it a smiley...
I'm not sure what to do here. I created an arbitrary form of q(t) and q'(t) and found a basis for each of these.
Now I'm lost, and I'm pretty certain I was never on the right track. Any help would be extremely appreciated.