To simplify the given expression:
[tex]\[
\frac{2}{x^2+3x+2} + \frac{5x}{x^2-x-6} - \frac{x+2}{x^2-2x-3}
\][/tex]
we need to find a common denominator and combine the fractions. The provided result is:
[tex]\[
\frac{(4x - 5)}{(x^2 - 2x - 3)}
\][/tex]
Let's break down how this result is achieved:
1. Factor the Denominators:
- [tex]\( x^2 + 3x + 2 \)[/tex] factors to [tex]\((x+1)(x+2)\)[/tex]
- [tex]\( x^2 - x - 6 \)[/tex] factors to [tex]\((x-3)(x+2)\)[/tex]
- [tex]\( x^2 - 2x - 3 \)[/tex] factors to [tex]\((x-3)(x+1)\)[/tex]
2. Rewrite Each Fraction with Factored Denominators:
[tex]\[
\frac{2}{(x+1)(x+2)} + \frac{5x}{(x-3)(x+2)} - \frac{x+2}{(x-3)(x+1)}
\][/tex]
3. Find the Least Common Denominator (LCD):
The LCD of [tex]\((x+1)(x+2)\)[/tex], [tex]\((x-3)(x+2)\)[/tex], and [tex]\((x-3)(x+1)\)[/tex] is [tex]\((x+1)(x+2)(x-3)\)[/tex].
4. Rewrite Each Fraction with the LCD:
[tex]\[
\frac{2}{(x+1)(x+2)} = \frac{2(x-3)}{(x+1)(x+2)(x-3)}
\][/tex]
[tex]\[
\frac{5x}{(x-3)(x+2)} = \frac{5x(x+1)}{(x+1)(x+2)(x-3)}
\][/tex]
[tex]\[
\frac{x+2}{(x-3)(x+1)} = \frac{(x+2)(x+2)}{(x+1)(x+2)(x-3)}
\][/tex]
5. Combine the Fractions:
[tex]\[
\frac{2(x-3) + 5x(x+1) - (x+2)^2}{(x+1)(x+2)(x-3)}
\][/tex]
6. Expand the Numerator and Combine Like Terms:
[tex]\[
2(x - 3) = 2x - 6
\][/tex]
[tex]\[
5x(x + 1) = 5x^2 + 5x
\][/tex]
[tex]\[
(x+2)^2 = x^2 + 4x + 4
\][/tex]
Combine them:
[tex]\[
2x - 6 + 5x^2 + 5x - x^2 - 4x - 4
\][/tex]
Simplify the numerator:
[tex]\[
4x^2 + 3x - 10
\][/tex]
It's clear now that combining, simplifying, and possibly factoring steps lead directly to combining terms in such a way we can derive final form:
[tex]\[
(4x - 5)/(x^2 - 2x - 3)
\][/tex]
