Sure, let's solve the problem step by step:
We are given the functions:
[tex]\[ f(x) = 3x + 5 \][/tex]
[tex]\[ g(x) = 4x^2 - 2 \][/tex]
[tex]\[ h(x) = x^2 - 3x + 1 \][/tex]
Our goal is to find [tex]\( f(x) - g(x) - h(x) \)[/tex].
1. Substitute the expressions for [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex]:
[tex]\[
f(x) - g(x) - h(x) = (3x + 5) - (4x^2 - 2) - (x^2 - 3x + 1)
\][/tex]
2. Distribute the negative signs:
[tex]\[
(3x + 5) - 4x^2 + 2 - x^2 + 3x - 1
\][/tex]
3. Combine like terms:
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
-4x^2 - x^2 = -5x^2
\][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[
3x + 3x = 6x
\][/tex]
- Combine the constant terms:
[tex]\[
5 + 2 - 1 = 6
\][/tex]
4. Put everything together:
[tex]\[
-5x^2 + 6x + 6
\][/tex]
So, the simplified result is:
[tex]\[ f(x) - g(x) - h(x) = -5x^2 + 6x + 6 \][/tex]
Therefore, the correct answer from the given choices is:
[tex]\[ \boxed{-5x^2 + 6x + 6} \][/tex]
