Find a fifthdegree polynomial that has a zero of multiplicit

Find a fifth-degree polynomial that has a zero of multiplicity 2 at x = 3, a zero at x = 5, and the factor x2 + 3x + 9.

zeros: x= 3 ( multiplicity 2)

x = 5

Factor : x^2 +3x+9

So, we can write the polynomial as:

f(x) = ( x-3)^2(x-5)(x^2 +3x+9)

= (x^2 +9 -6x)(x-5)(x^2 +3x +9)

= (x^3 -5x^2 +9x -45 -6x^2 +30x)(x^2 +3x +9)

=(x^3 -11x^2 +39x -45)(x^2 +3x+9)

= x^5 +3x^4 +9x^3 -11x^4 -33x^3 -99x^2 +39x^3 + 117x^2 -45x^2 -135x -405

=x^5 -8x^4+ 15x^3 -27x^2 - 216x -405