To find the sum of the rational expressions [tex]\(\frac{3x}{x+9} + \frac{x}{x-4}\)[/tex], we need to combine them into a single fraction. Here’s the step-by-step solution:
1. Identify the Least Common Denominator (LCD):
The denominators of the given expressions are [tex]\(x+9\)[/tex] and [tex]\(x-4\)[/tex]. The least common denominator (LCD) of these two denominators is [tex]\((x+9)(x-4)\)[/tex].
2. Rewrite each fraction with the LCD as the denominator:
For [tex]\(\frac{3x}{x+9}\)[/tex]:
[tex]\[
\frac{3x}{x+9} \cdot \frac{x-4}{x-4} = \frac{3x(x-4)}{(x+9)(x-4)}
\][/tex]
For [tex]\(\frac{x}{x-4}\)[/tex]:
[tex]\[
\frac{x}{x-4} \cdot \frac{x+9}{x+9} = \frac{x(x+9)}{(x+9)(x-4)}
\][/tex]
3. Combine the two fractions:
Now that both fractions have the same denominator, we can add the numerators:
[tex]\[
\frac{3x(x-4)}{(x+9)(x-4)} + \frac{x(x+9)}{(x+9)(x-4)} = \frac{3x(x-4) + x(x+9)}{(x+9)(x-4)}
\][/tex]
4. Simplify the numerator:
Expand the expressions in the numerator:
[tex]\[
3x(x-4) + x(x+9) = 3x^2 - 12x + x^2 + 9x = 4x^2 - 3x
\][/tex]
5. Write the simplified combined fraction:
The result is:
[tex]\[
\frac{4x^2 - 3x}{(x+9)(x-4)}
\][/tex]
Recognize that [tex]\((x+9)(x-4)\)[/tex] can be rewritten as [tex]\(x^2 + 5x - 36\)[/tex].
6. Final simplified expression:
Thus, the sum of the given rational expressions is:
[tex]\[
\frac{4x^2 - 3x}{x^2 + 5x - 36}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{4x^2-3x}{x^2+5x-36}}
\][/tex]
So the correct option is [tex]\(C\)[/tex].
