To solve this problem, we need to identify the correct expression for [tex]\((f-g)(x)\)[/tex] from the given options. The expressions given are:
1. [tex]\(x^2 + x - 4\)[/tex]
2. [tex]\(x^2 + x + 4\)[/tex]
3. [tex]\(x^2 - x + 6\)[/tex]
4. [tex]\(x^2 + x + 6\)[/tex]
Given that [tex]\((f-g)(x) = x^2 + x - 4\)[/tex], we need to determine which of the expressions matches this result.
Let's examine each option step-by-step:
1. Check the first option [tex]\(x^2 + x - 4\)[/tex]:
[tex]\[
x^2 + x - 4
\][/tex]
This exactly matches the given [tex]\((f-g)(x)\)[/tex].
2. Check the second option [tex]\(x^2 + x + 4\)[/tex]:
[tex]\[
x^2 + x + 4
\][/tex]
This does not match the given [tex]\(x^2 + x - 4\)[/tex].
3. Check the third option [tex]\(x^2 - x + 6\)[/tex]:
[tex]\[
x^2 - x + 6
\][/tex]
This does not match the given [tex]\(x^2 + x - 4\)[/tex].
4. Check the fourth option [tex]\(x^2 + x + 6\)[/tex]:
[tex]\[
x^2 + x + 6
\][/tex]
This does not match the given [tex]\(x^2 + x - 4\)[/tex].
From the comparison, the first expression [tex]\(x^2 + x - 4\)[/tex] is the only one that matches the given [tex]\((f-g)(x)\)[/tex].
Therefore, the correct expression for [tex]\((f-g)(x)\)[/tex] is the first one, and its index is [tex]\(0\)[/tex].
