You are given a 4th degree polynomial and a root of multiplicity two. This means your polynomial has the form (x-2)^2*(x^2 + bx + c) = 0.
You have factorized the polynomial wrong because you have (x-2)^2 * (x-3) * a quadratic which will give you a fifth degree polynomial.
If 2 and 3 are the only two zeroes then x^4 - 4x^3 - 5x^2 + 36x - 36 = (x-2)^2*(x-3)^2 and an expansion should verify this if it is the case.
Also try dividing your polynomial by (x-2)^2 or x^2 - 4x + 4 and see what the quotient is (the remainder should be 0).