By inspection x=1 is a root, so divide by this root:
1 | 3 -11 15 -9 2
3 3 -8 7 -2
3 -8 7 -2 | 0 = 3x^3-8x^2+7x-2, which, by inspection, also has x=1 as a root, so divide again:
1 | 3 -8 7 -2
3 3 -5 2
3 -5 2 | 0 = 3x^2-5x+2=(x-1)(3x-2).
The roots are 1 (three times) and 2/3: factorisation is (x-1)^3(3x-2).