You are on the right track.

There was be an easier way that I have forgotten however this will solve it.

Use the base frame and the derivative of the base frame.

First work out f(0)=2 and f'(0)=-3. (Ie substitute x for 0)

You will get two equations.

f(0)=a0^3+b0^2+c0+d=2

This gives d=2

And c=-3 from the derviative.

Now you do it for the two zeros. Ie f(-1)=0 and f(2)=0

This will give two equations with a,b, c and d. So you can the sustitue what c and d are into them (since you already know them) and then solve two simulataneous equations.

That is the method. You just need to do it now

Let me know if you are still stuck.

And your notation is interesting.