To begin this question I would first rewrite into a non-homogenous solution to find out what case I will be working with. Once I found my case I would then move on to step 2 and that is writing out my partial solution. Then I would begin to find Yp, in this solution just divide the roots on both sides to get the -24e^-2t over the roots, then plug in the variable D with the exponent -2 and solve to get the Yp value. Once that is completed all we have to do is put the solution together. The general solution is always y(t) = Yh(t) + Yp(t). That would then conclude my answer for all the parts within this question.
I had that same problem, i pretty much had the same process as you, though you were able to convey the same information by writing less stuff than i did. Other differences being that i used A and B instead of C1 and C2.
Good that you were able to solve this using a method from a different course. I also had a bit of trouble in case 1 but was also able to figure it out after identifying the error.