To find the oblique asymptote of the function [tex]\( g(x) = \frac{x^2 - 3x - 5}{x + 2} \)[/tex], we need to perform polynomial long division.
Step-by-Step Solution:
1. Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
2. Multiply the entire divisor [tex]\((x+2)\)[/tex] by [tex]\(x\)[/tex] and subtract this from the original numerator [tex]\((x^2 - 3x - 5)\)[/tex]:
[tex]\[
(x^2 - 3x - 5) - (x \cdot (x + 2)) = (x^2 - 3x - 5) - (x^2 + 2x)
\][/tex]
Simplify this:
[tex]\[
(x^2 - 3x - 5) - (x^2 + 2x) = -5x - 5
\][/tex]
3. Divide the next term of the result (-5x) by the leading term of the divisor (x):
[tex]\[
\frac{-5x}{x} = -5
\][/tex]
4. Multiply the entire divisor [tex]\((x + 2)\)[/tex] by [tex]\(-5\)[/tex] and subtract:
[tex]\[
(-5x - 5) - (-5 \cdot (x + 2)) = (-5x - 5) - (-5x - 10)
\][/tex]
Simplify this:
[tex]\[
(-5x - 5) - (-5x - 10) = 5
\][/tex]
Our quotient from the division process is [tex]\(x - 5\)[/tex] and the remainder is [tex]\(5\)[/tex].
Since we are only interested in the oblique asymptote (ignoring the remainder as [tex]\(x \to \infty\)[/tex]), we identify the oblique asymptote by the quotient we obtained:
[tex]\[
y = x - 5
\][/tex]
Therefore, the oblique asymptote of [tex]\( g(x) = \frac{x^2 - 3x - 5}{x + 2} \)[/tex] is:
[tex]\( y = x - 5 \)[/tex]
From the given choices, the correct answer is [tex]\( y = x - 5 \)[/tex].
