Sure, let's find the sum of the two fractions step by step.
Given the expression:
[tex]\[
\frac{3}{x^2 - 9} + \frac{5}{x + h}
\][/tex]
First, observe the denominators. Notice that [tex]\(x^2 - 9\)[/tex] can be factored.
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
So, the first fraction becomes:
[tex]\[
\frac{3}{(x - 3)(x + 3)}
\][/tex]
We seek a common denominator for the two fractions. The denominators are [tex]\((x - 3)(x + 3)\)[/tex] and [tex]\(x + h\)[/tex]. The least common denominator (LCD) will be the product of these distinct linear factors:
[tex]\[
\text{LCD} = (x - 3)(x + 3)(x + h)
\][/tex]
Next, we rewrite each fraction with the LCD as the combined denominator.
Rewrite the first fraction:
[tex]\[
\frac{3}{(x - 3)(x + 3)} = \frac{3(x + h)}{(x - 3)(x + 3)(x + h)}
\][/tex]
Rewrite the second fraction:
[tex]\[
\frac{5}{x + h} = \frac{5(x - 3)(x + 3)}{(x + h)(x - 3)(x + 3)}
\][/tex]
Now, add the two fractions with their new denominators:
[tex]\[
\frac{3(x + h)}{(x - 3)(x + 3)(x + h)} + \frac{5(x - 3)(x + 3)}{(x - 3)(x + 3)(x + h)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{3(x + h) + 5(x - 3)(x + 3)}{(x - 3)(x + 3)(x + h)}
\][/tex]
Simplify the numerator:
[tex]\[
3(x + h) + 5(x^2 - 9)
\][/tex]
[tex]\[3(x + h) + 5(x^2 - 9) = 3x + 3h + 5x^2 - 45\][/tex]
Finally, the sum of the fractions is:
[tex]\[
\frac{3}{x^2 - 9} + \frac{5}{x + h} = \frac{3(x + h) + 5(x^2 - 9)}{(x - 3)(x + 3)(x + h)} = \frac{3 + 5(x - 3)(x + 3)}{(x - 3)(x + 3)(x + h)}
\][/tex]
Thus, the sum of the fractions is:
[tex]\[
\boxed{\frac{3}{x^2 - 9} + \frac{5}{x + h}}
\][/tex]
