To determine which of the given options fits the expression [tex]\((f - g)(x) = x^2 + x + k\)[/tex] for some constant [tex]\(k\)[/tex], we need to determine which polynomial can be expressed in the form [tex]\( x^2 + x + k \)[/tex].
The provided options 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]
We need the polynomial [tex]\((f - g)(x)\)[/tex] to match the form [tex]\( x^2 + x + k \)[/tex].
Let's analyze each option:
1. Option 1: [tex]\( x^2 + x - 4 \)[/tex]
This polynomial fits the form [tex]\( x^2 + x + k \)[/tex] where [tex]\( k = -4 \)[/tex].
2. Option 2: [tex]\( x^2 + x + 4 \)[/tex]
This polynomial fits the form [tex]\( x^2 + x + k \)[/tex] where [tex]\( k = +4 \)[/tex].
3. Option 3: [tex]\( x^2 - x + 6 \)[/tex]
This polynomial does not fit the form [tex]\( x^2 + x + k \)[/tex] because the coefficient of [tex]\( x \)[/tex] is [tex]\(-1\)[/tex] instead of [tex]\(+1\)[/tex].
4. Option 4: [tex]\( x^2 + x + 6 \)[/tex]
This polynomial fits the form [tex]\( x^2 + x + k \)[/tex] where [tex]\( k = +6 \)[/tex].
Given that we need [tex]\((f - g)(x)\)[/tex] to be of the form [tex]\( x^2 + x + k \)[/tex], the options that fit this form are:
- Option 1: [tex]\( x^2 + x - 4 \)[/tex]
- Option 2: [tex]\( x^2 + x + 4 \)[/tex]
- Option 4: [tex]\( x^2 + x + 6 \)[/tex]
Among these options, we must ensure that we are picking the correct constant value [tex]\(k\)[/tex] to match the polynomial given in the problem.
The correct polynomial [tex]\(x^2 + x + k \)[/tex] that fits the typical required form given the problem's context is:
Option 4: [tex]\(x^2 + x + 6\)[/tex]
Thus, the correct answer is:
[tex]\[ (f-g)(x) = x^2 + x + 6 \][/tex]
