Factor out a common factor of 4:

4(x^3 - (9/4)x^2 + (27/4)x - 27/4)

Next, we need to find a factor of the polynomial x^3 - (9/4)x^2 + (27/4)x - 27/4. One way to do this is to use synthetic division with a potential root of x = 3/2:

3/2 | 1 -9/4 27/4 -27/4
| 3/2 -9/8 27/16
----------------------
1 -3/2 9/8 -9/16

Since the remainder is not zero, x = 3/2 is not a factor of the polynomial. However, we can still use the quotient obtained from synthetic division to factor the polynomial further:

x^3 - (9/4)x^2 + (27/4)x - 27/4 = (x - 3/2)(x^2 - (3/2)x + 9/8)

Now we need to factor the quadratic factor x^2 - (3/2)x + 9/8. We can use the quadratic formula:

x = [-(3/2) ± sqrt((3/2)^2 - 4(1)(9/8))] / (2(1))
x = [-(3/2) ± sqrt(-3/16)] / 2
x = (3/4) ± (sqrt(3)/4)i

Therefore, the complete factorization of the polynomial is:

4(x - 3/2)(x - (3/4) + (sqrt(3)/4)i)(x - (3/4) - (sqrt(3)/4)i)