To subtract the given rational expressions [tex]\( \frac{2x + 1}{x - 4} - \frac{x - 5}{x^2 - 3x - 4} \)[/tex], we'll follow these steps:
1. Factor the denominator [tex]\( x^2 - 3x - 4 \)[/tex]:
[tex]\[
x^2 - 3x - 4 = (x - 4)(x + 1)
\][/tex]
This simplifies our second expression to:
[tex]\[
\frac{x - 5}{x^2 - 3x - 4} = \frac{x - 5}{(x - 4)(x + 1)}
\][/tex]
2. Rewrite both fractions with a common denominator:
The common denominator is [tex]\( (x - 4)(x + 1) \)[/tex]. Thus, rewrite the first expression:
[tex]\[
\frac{2x + 1}{x - 4} = \frac{(2x + 1)(x + 1)}{(x - 4)(x + 1)}
\][/tex]
So, we have:
[tex]\[
\frac{(2x + 1)(x + 1)}{(x - 4)(x + 1)} - \frac{x - 5}{(x - 4)(x + 1)}
\][/tex]
3. Expand and simplify the numerators before subtraction:
Expand [tex]\( (2x + 1)(x + 1) \)[/tex]:
[tex]\[
(2x + 1)(x + 1) = 2x^2 + 2x + x + 1 = 2x^2 + 3x + 1
\][/tex]
So, we have:
[tex]\[
\frac{2x^2 + 3x + 1}{(x - 4)(x + 1)} - \frac{x - 5}{(x - 4)(x + 1)}
\][/tex]
4. Combine the fractions into a single fraction:
[tex]\[
\frac{2x^2 + 3x + 1 - (x - 5)}{(x - 4)(x + 1)}
\][/tex]
5. Simplify the numerator:
Simplify [tex]\( 2x^2 + 3x + 1 - (x - 5) \)[/tex]:
[tex]\[
2x^2 + 3x + 1 - x + 5 = 2x^2 + 2x + 6
\][/tex]
6. Final simplified form:
[tex]\[
\frac{2x^2 + 2x + 6}{(x - 4)(x + 1)} = \frac{2(x^2 + x + 3)}{(x - 4)(x + 1)}
\][/tex]
Thus, the simplified result of the subtraction [tex]\( \frac{2x + 1}{x - 4} - \frac{x - 5}{x^2 - 3x - 4} \)[/tex] is:
[tex]\[
\frac{2(x^2 + x + 3)}{(x - 4)(x + 1)}
\][/tex]
Using the choices provided, the correct answer is:
[tex]\(\frac{2(x^2 + x + 3)}{(x - 4)(x + 1)}\)[/tex].
