x = 1 is not a solution. You can easily prove that yourself.
When x = 2,
4x^3 + 8x^2 − 9x − 18 = 28
So that is not a solution, either. Neither is x = 3. How did you come up with your answers?
When x = -2,
-32 +32 +18 -18 = 0
So x+2 is a factor of the polynomial.
The other factor is
(4x^3 + 8x^2 − 9x − 18)/(x+2)
= (4x^2 -9) = (2x +3)(2x-3)
So the other two solutions are
x = 3/2 and x = -3/2
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