To solve the problem, which involves finding the difference between the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], let's proceed with the following steps:
1. Write down the expressions for [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = 3x^2 - 2x - 9
\][/tex]
[tex]\[
g(x) = 3x^3 + 2x^2 - 4x - 9
\][/tex]
2. Find [tex]\( (f - g)(x) \)[/tex], which is [tex]\( f(x) - g(x) \)[/tex]:
[tex]\[
f(x) - g(x) = (3x^2 - 2x - 9) - (3x^3 + 2x^2 - 4x - 9)
\][/tex]
3. Distribute the negative sign in front of [tex]\( g(x) \)[/tex] and combine like terms:
[tex]\[
f(x) - g(x) = 3x^2 - 2x - 9 - 3x^3 - 2x^2 + 4x + 9
\][/tex]
4. Combine like terms:
[tex]\[
f(x) - g(x) = -3x^3 + 3x^2 - 2x^2 - 2x + 4x - 9 + 9
\][/tex]
[tex]\[
f(x) - g(x) = -3x^3 + (3x^2 - 2x^2) + (-2x + 4x) + (-9 + 9)
\][/tex]
[tex]\[
f(x) - g(x) = -3x^3 + x^2 + 2x
\][/tex]
5. State the final result of [tex]\( (f - g)(x) \)[/tex]:
[tex]\[
(f - g)(x) = -3x^3 + x^2 + 2x
\][/tex]
Therefore, the correct result of [tex]\( (f - g)(x) \)[/tex] is:
[tex]\[
\boxed{x(-3x^2 + x + 2)}
\][/tex]
Among the provided options, this expression matches with:
[tex]\[
\boxed{-3x^3 + x^2 + 2x}
\][/tex]
