To find \((f - g)(x)\) for the given functions \(f(x)\) and \(g(x)\), we perform the following steps:
1. Write down the given functions:
[tex]\[
f(x) = 2x^2 - 5
\][/tex]
[tex]\[
g(x) = x^2 - 4x - 8
\][/tex]
2. Subtract \(g(x)\) from \(f(x)\):
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
Substituting the expressions for \(f(x)\) and \(g(x)\), we get:
[tex]\[
(f - g)(x) = (2x^2 - 5) - (x^2 - 4x - 8)
\][/tex]
3. Distribute the negative sign and combine like terms:
[tex]\[
(f - g)(x) = 2x^2 - 5 - x^2 + 4x + 8
\][/tex]
4. Simplify the expression by combining like terms:
- Combine the \(x^2\) terms:
[tex]\[
2x^2 - x^2 = x^2
\][/tex]
- The \(x\) terms remain as is:
[tex]\[
+4x
\][/tex]
- Combine the constant terms:
[tex]\[
-5 + 8 = 3
\][/tex]
So, the simplified expression is:
[tex]\[
(f - g)(x) = x^2 + 4x + 3
\][/tex]
5. Check the multiple-choice options:
- A. \(x^2 - 4x - 3\)
- B. \(x^2 + 4x + 3\)
- C. \(-x^2 - 13\)
- D. \(3x^2 - 4x - 13\)
The correct expression \((f - g)(x)\) which we found is \(x^2 + 4x + 3\).
Thus, the correct answer is:
[tex]\[
\boxed{B}
\][/tex]
