Sure! Let's simplify the given expression step-by-step.
Given the expression:
[tex]\[
\frac{2n - 9}{x^2 - x - 2}
\][/tex]
The first step is to factor the denominator, [tex]\(x^2 - x - 2\)[/tex].
We can factor [tex]\(x^2 - x - 2\)[/tex] using the methods of factoring quadratic equations. We look for two numbers that multiply to [tex]\(-2\)[/tex] and add to [tex]\(-1\)[/tex].
The factors of [tex]\(-2\)[/tex] that add up to [tex]\(-1\)[/tex] are [tex]\(-2\)[/tex] and [tex]\(1\)[/tex]. So, we can write:
[tex]\[
x^2 - x - 2 = (x - 2)(x + 1)
\][/tex]
So, the expression now is:
[tex]\[
\frac{2n - 9}{(x - 2)(x + 1)}
\][/tex]
There are no common factors in the numerator [tex]\(2n - 9\)[/tex] and the denominator [tex]\((x - 2)(x + 1)\)[/tex] that can be canceled out directly. Hence, we simplify this expression by rewriting it as:
[tex]\[
\frac{2n - 9}{(x - 2)(x + 1)} = \frac{9 - 2n}{-x^2 + x + 2}
\][/tex]
While we have rewritten the expression keeping the right form of numerator relationship, so as per the symbolic reduction, the equivalent simplest form is retained.
Thus, the simplified expression is:
[tex]\[
\frac{9 - 2n}{-x^2 + x + 2}
\][/tex]
