To find [tex]\((f - g)(x)\)[/tex] where [tex]\(f(x) = 5x^2 - 3\)[/tex] and [tex]\(g(x) = x^2 - 4x - 8\)[/tex], we'll proceed as follows:
1. Express the functions:
- [tex]\(f(x) = 5x^2 - 3\)[/tex]
- [tex]\(g(x) = x^2 - 4x - 8\)[/tex]
2. Subtract [tex]\(g(x)\)[/tex] from [tex]\(f(x)\)[/tex]:
[tex]\[(f - g)(x) = f(x) - g(x)\][/tex]
3. Perform the subtraction:
[tex]\[
(f - g)(x) = (5x^2 - 3) - (x^2 - 4x - 8)
\][/tex]
4. Distribute the negative sign:
[tex]\[
(f - g)(x) = 5x^2 - 3 - x^2 + 4x + 8
\][/tex]
5. Combine like terms:
- For [tex]\(x^2\)[/tex] terms: [tex]\(5x^2 - x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(4x\)[/tex]
- For constant terms: [tex]\(-3 + 8 = 5\)[/tex]
Therefore,
[tex]\[
(f - g)(x) = 4x^2 + 4x + 5
\][/tex]
So, the correct answer is:
[tex]\[ B. (f - g)(x) = 4x^2 + 4x + 5 \][/tex]
Hence, (f - g)(x) = 4x^2 + 4x + 5.
