To solve the expression [tex]\(\frac{x^2 + 5x + 6}{x^2 - 4} + \frac{4x - 18}{2x - 4}\)[/tex], let's go through the steps to simplify each fraction first and then add them.
1. Simplification of [tex]\(\frac{x^2 + 5x + 6}{x^2 - 4}\)[/tex]:
[tex]\[
\frac{x^2 + 5x + 6}{x^2 - 4}
\][/tex]
Factorize the numerator and the denominator:
- Numerator: [tex]\(x^2 + 5x + 6 = (x + 2)(x + 3)\)[/tex]
- Denominator: [tex]\(x^2 - 4 = (x - 2)(x + 2)\)[/tex]
So, the fraction becomes:
[tex]\[
\frac{(x + 2)(x + 3)}{(x - 2)(x + 2)}
\][/tex]
Cancel out the common factor [tex]\((x + 2)\)[/tex]:
[tex]\[
\frac{x + 3}{x - 2}
\][/tex]
2. Simplification of [tex]\(\frac{4x - 18}{2x - 4}\)[/tex]:
[tex]\[
\frac{4x - 18}{2x - 4}
\][/tex]
We can factor out the greatest common factor in both the numerator and the denominator:
- Numerator: [tex]\(4x - 18 = 2(2x - 9)\)[/tex]
- Denominator: [tex]\(2x - 4 = 2(x - 2)\)[/tex]
So, the fraction becomes:
[tex]\[
\frac{2(2x - 9)}{2(x - 2)}
\][/tex]
Cancel out the common factor [tex]\(2\)[/tex]:
[tex]\[
\frac{2x - 9}{x - 2}
\][/tex]
3. Addition of the simplified fractions [tex]\(\frac{x + 3}{x - 2} + \frac{2x - 9}{x - 2}\)[/tex]:
Since both fractions have the same denominator [tex]\((x - 2)\)[/tex], we can add them directly:
[tex]\[
\frac{x + 3}{x - 2} + \frac{2x - 9}{x - 2} = \frac{(x + 3) + (2x - 9)}{x - 2}
\][/tex]
Combine the numerators:
[tex]\[
\frac{x + 3 + 2x - 9}{x - 2} = \frac{3x - 6}{x - 2}
\][/tex]
Factor out the greatest common factor in the numerator:
[tex]\[
\frac{3(x - 2)}{x - 2}
\][/tex]
Cancel out the common factor [tex]\((x - 2)\)[/tex]:
[tex]\[
3
\][/tex]
So, the result of the given expression [tex]\(\frac{x^2 + 5x + 6}{x^2 - 4} + \frac{4x - 18}{2x - 4}\)[/tex] simplifies to:
[tex]\[
\boxed{3}
\][/tex]
