2x⁵-11x⁴+69x³+135x²-675x=
x(2x⁴-11x³+69x²+135x-675).
Since 3-6i is a zero, then so is 3+6i.
If we combine them we get (x-3+6i)(x-3-6i)=x²-6x+9+36=
x²-6x+45 as a factor of the quartic in parentheses.
x²-6x+45 ) 2x⁴-11x³+69x²+135x-675 | 2x²+x-15
2x⁴-12x³+ 90x²
x³- 21x²+135x
x³- 6x²+ 45x
-15x²+ 90x-675
We can find the zeroes of 2x²+x-15=(2x-5)(x+3). So the remaining zeroes are 5/2 and -3.
The zeroes are therefore 0, 3-6i, 3+6i, 5/2, -3. The only complex factors are 3-6i and 3+6i.
Factored into real factors we have x(2x-5)(x+3)(x²-6x+45).