To solve the problem of subtracting the function [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex], we need to follow these steps:
1. Write down the functions:
[tex]\[
f(x) = -5x^2 + x - 2
\][/tex]
[tex]\[
g(x) = -3x^2 + 3x + 9
\][/tex]
2. Subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) - g(x) = (-5x^2 + x - 2) - (-3x^2 + 3x + 9)
\][/tex]
3. Distribute the negative sign across [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) - g(x) = -5x^2 + x - 2 - (-3x^2 + 3x + 9)
\][/tex]
[tex]\[
f(x) - g(x) = -5x^2 + x - 2 + 3x^2 - 3x - 9
\][/tex]
4. Combine like terms:
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
-5x^2 + 3x^2 = -2x^2
\][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[
x - 3x = -2x
\][/tex]
- Combine the constant terms:
[tex]\[
-2 - 9 = -11
\][/tex]
5. Write down the final result:
[tex]\[
f(x) - g(x) = -2x^2 - 2x - 11
\][/tex]
The correct choice is:
[tex]\[
\boxed{-2 x^2-2 x-11}
\][/tex]
Thus, the correct answer is (b) [tex]\( -2 x^2 - 2 x - 11 \)[/tex].
