To find [tex]\((f - g)(x)\)[/tex], we need to subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]. Let's write down both functions and then perform the subtraction step-by-step.
We are given:
[tex]\[ f(x) = 3x^3 + 4x^2 - 8x - 2 \][/tex]
[tex]\[ g(x) = 3x - 5 \][/tex]
The expression for [tex]\((f - g)(x)\)[/tex] is:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
To find this, we substitute the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f - g)(x) = (3x^3 + 4x^2 - 8x - 2) - (3x - 5) \][/tex]
Next, we distribute the minus sign across [tex]\(g(x)\)[/tex]:
[tex]\[ (f - g)(x) = 3x^3 + 4x^2 - 8x - 2 - 3x + 5 \][/tex]
Now, we combine like terms:
[tex]\[
\begin{align*}
3x^3 & \quad \text{(there is no other } x^3 \text{ term)} \\
+ 4x^2 & \quad \text{(there is no other } x^2 \text{ term)} \\
- 8x - 3x & = -11x \\
- 2 + 5 & = 3 \\
\end{align*}
\][/tex]
Putting it all together:
[tex]\[ (f - g)(x) = 3x^3 + 4x^2 - 11x + 3 \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{(f-g)(x)=3x^3+4x^2-11x+3} \][/tex]
So the correct answer is:
[tex]\[ \text{C. } (f-g)(x)=3 x^3+4 x^2-11 x+3 \][/tex]
