Why do you not use the information you are given?
And since p has degree six, any other factor must be linear:zero of multiplicity 2 at t =1 implies (t-1)^2 is a factor
zero of multiplicity 3 at t = 2 implies (t-2)^3 is a factor
p(t)= (t- 1)^2(t- 2)^3(at+ b)
Now put the other information into that.