To find \((f - g)(x)\), we need to subtract \(g(x)\) from \(f(x)\).
Given: \[ f(x) = x^2 - 3x + 9 \] \[ g(x) = 3x^3 + 2x^2 - 4x - 9 \]
Now, let's compute \((f - g)(x)\):
\[ (f - g)(x) = f(x) - g(x) \] \[ = (x^2 - 3x + 9) - (3x^3 + 2x^2 - 4x - 9) \]
Distributing the negative sign to \(g(x)\): \[ = x^2 - 3x + 9 - 3x^3 - 2x^2 + 4x + 9 \]
Now, combine like terms:
- The \(x^3\) term: \(-3x^3\)
- The \(x^2\) terms: \(x^2 - 2x^2 = -x^2\)
- The \(x\) terms: \(-3x + 4x = x\)
- The constant terms: \(9 + 9 = 18\)
Putting it all together: \[ (f - g)(x) = -3x^3 - x^2 + x + 18 \]
Thus, the answer corresponds to option: \[ \boxed{-3x^3 - x^2 + x + 18} \]