Given two roots we have (x-√3)(x+√3)=x2-3 is a factor of the polynomial. Divide by this factor:
x2 + x - 6
x2-3 ) x4 + x3 - 9x2 - 3x + 18
x4 - 3x2
x3 - 6x2 - 3x + 18
x3 - 3x
- 6x2 + 18
- 6x2 + 18
0
The quotient factorises: (x+3)(x-2), so the zeroes are -3, -√3, √3, 2.