To solve this problem, we need to perform the subtraction of the two rational expressions and simplify the result. Let's break the problem into manageable steps:
1. Factor the denominators:
- Factor the first denominator [tex]\( x^2 - x - 2 \)[/tex]:
[tex]\[
x^2 - x - 2 = (x - 2)(x + 1)
\][/tex]
- Factor the second denominator [tex]\( x^2 + 7x - 18 \)[/tex]:
[tex]\[
x^2 + 7x - 18 = (x - 2)(x + 9)
\][/tex]
2. Find the common denominator:
The common denominator is the least common multiple (LCM) of the factored denominators:
[tex]\[
(x - 2)(x + 1) \text{ and } (x - 2)(x + 9)
\][/tex]
The LCM is:
[tex]\[
(x - 2)(x + 1)(x + 9)
\][/tex]
3. Rewrite each fraction with the common denominator:
- For the first fraction:
[tex]\[
\frac{x+9}{(x-2)(x+1)}
\][/tex]
Multiply numerator and denominator by [tex]\( (x + 9) \)[/tex]:
[tex]\[
\frac{(x + 9)(x + 9)}{(x-2)(x+1)(x+9)} = \frac{(x + 9)^2}{(x-2)(x+1)(x+9)}
\][/tex]
- For the second fraction:
[tex]\[
\frac{6x + 5}{(x-2)(x+9)}
\][/tex]
Multiply numerator and denominator by [tex]\( (x + 1) \)[/tex]:
[tex]\[
\frac{(6x + 5)(x + 1)}{(x-2)(x+1)(x+9)}
\][/tex]
4. Subtract the fractions:
[tex]\[
\frac{(x + 9)^2}{(x-2)(x+1)(x+9)} - \frac{(6x + 5)(x + 1)}{(x-2)(x+1)(x+9)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{(x + 9)^2 - (6x + 5)(x + 1)}{(x-2)(x+1)(x+9)}
\][/tex]
5. Simplify the numerator:
[tex]\[
(x + 9)^2 - (6x + 5)(x + 1)
\][/tex]
Expand both expressions:
[tex]\[
(x + 9)^2 = x^2 + 18x + 81
\][/tex]
[tex]\[
(6x + 5)(x + 1) = 6x^2 + 11x + 5
\][/tex]
Subtract the expanded expressions:
[tex]\[
(x^2 + 18x + 81) - (6x^2 + 11x + 5) = x^2 + 18x + 81 - 6x^2 - 11x - 5
\][/tex]
Combine like terms:
[tex]\[
x^2 + 18x + 81 - 6x^2 - 11x - 5 = -5x^2 + 7x + 76
\][/tex]
6. Write the final expression:
[tex]\[
\frac{-5x^2 + 7x + 76}{(x-2)(x+1)(x+9)}
\][/tex]
So the simplified subtraction of the given fractions is:
[tex]\[
\frac{-5x^2 + 7x + 76}{(x-2)(x+1)(x+9)}
\][/tex]
