Recap: You know that s(t) is a cubic because when you take two derivatives of a cubic you get At + B which is the same form as a(t) = 24t + 6.
So using the initial conditions eg. s(0) = 0, the a(t) formula, s(t) = q*t^3 + b*t^2 + c*t + d, and knowing that s' = v and s'' = a, solve for q,b,c...