To find the sum of [tex]\(\frac{7x}{x^2 - 4}\)[/tex] and [tex]\(\frac{2}{x + 2}\)[/tex], we can follow these steps:
1. Simplify the Denominators:
- Notice that [tex]\(x^2 - 4\)[/tex] is a difference of squares and can be factored:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
2. Rewrite the Expressions:
- The first fraction is already expressed with the factored denominator:
[tex]\[
\frac{7x}{x^2 - 4} = \frac{7x}{(x - 2)(x + 2)}
\][/tex]
- Rewrite [tex]\(\frac{2}{x + 2}\)[/tex] over the common denominator [tex]\((x - 2)(x + 2)\)[/tex]:
[tex]\[
\frac{2}{x + 2} = \frac{2(x - 2)}{(x + 2)(x - 2)} = \frac{2x - 4}{(x + 2)(x - 2)}
\][/tex]
3. Add the Fractions:
- Now, both fractions have the same denominator [tex]\((x - 2)(x + 2)\)[/tex]. We can add the numerators:
[tex]\[
\frac{7x}{(x - 2)(x + 2)} + \frac{2x - 4}{(x - 2)(x + 2)} = \frac{7x + (2x - 4)}{(x - 2)(x + 2)}
\][/tex]
- Combine the terms in the numerator:
[tex]\[
\frac{7x + 2x - 4}{(x - 2)(x + 2)} = \frac{9x - 4}{(x - 2)(x + 2)}
\][/tex]
4. Simplified Result:
- The final simplified expression is:
[tex]\[
\frac{9x - 4}{x^2 - 4}
\][/tex]
Thus, the sum of [tex]\(\frac{7x}{x^2 - 4}\)[/tex] and [tex]\(\frac{2}{x + 2}\)[/tex] is:
[tex]\[
\boxed{\frac{9x - 4}{x^2 - 4}}
\][/tex]
