To find [tex]\( (f-g)(x) \)[/tex], we need to simplify the expression resulting from subtracting [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex].
Let's break this down step by step.
1. Consider the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- Assume [tex]\( f(x) = x^2 + x + a \)[/tex]
- Assume [tex]\( g(x) = x^2 + x + b \)[/tex]
2. Subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
(f - g)(x) = f(x) - g(x)
= (x^2 + x + a) - (x^2 + x + b)
\][/tex]
3. Simplify the expression:
[tex]\[
(f - g)(x) = (x^2 + x + a) - (x^2 + x + b)
= x^2 + x + a - x^2 - x - b
\][/tex]
Notice that [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex] terms cancel out:
[tex]\[
= a - b
\][/tex]
To match this with the given options:
1. Evaluate the given options:
- Option 1: [tex]\((x^2 + x - 4)\)[/tex]
- Option 2: [tex]\((x^2 + x + 4)\)[/tex]
- Option 3: [tex]\((x^2 - x + 6)\)[/tex]
- Option 4: [tex]\((x^2 + x + 6)\)[/tex]
Comparing our simplified expression [tex]\( a - b \)[/tex] with the options, the option that satisfies the criteria given our algebraic operations is:
[tex]\[
\boxed{x^2 + x + 6}
\][/tex]
So the correct answer is the fourth option: [tex]\((x^2 + x + 6)\)[/tex].
