To find \( (f \cdot g)(x) \), we first need to multiply the functions \( f(x) \) and \( g(x) \).
Given:
- \( f(x) = x^2 + 8x + 15 \)
- \( g(x) = \frac{5}{x^2 - 9} \)
Now, we calculate \( (f \cdot g)(x) \):
\[ (f \cdot g)(x) = f(x) \cdot g(x) = (x^2 + 8x + 15) \cdot \left(\frac{5}{x^2 - 9}\right) \]
Simplifying this:
\[ (f \cdot g)(x) = \frac{5(x^2 + 8x + 15)}{x^2 - 9} \]
Next, we expand the numerator:
\[ 5(x^2 + 8x + 15) = 5x^2 + 40x + 75 \]
Therefore:
\[ (f \cdot g)(x) = \frac{5x^2 + 40x + 75}{x^2 - 9} \]
The denominator, \( x^2 - 9 \), can be factored into \( (x - 3)(x + 3) \), but we will leave it as \( x^2 - 9 \) for now.
Now we have:
\[ (f \cdot g)(x) = \frac{5x^2 + 40x + 75}{x^2 - 9} \]
Next, let's check the options provided:
- \( \frac{5x + 25}{x - 3} \)
- \( \frac{5}{x^4 + 16x^3 + 94x^2 + 240x + 216} \)
- \( \frac{5x + 25}{x + 3} \)
- \( \frac{25}{x^4 - 18x^2 + 81} + \frac{40}{x^2 - 9} + 15 \)
None of the options directly appear to match our derived formula exactly.
However, we can inspect our formula and look if it can be simplified or altered to match one of the forms by examining purely rational functions or analyzing the possible equivalent forms.
Given the lack of straightforward results and in reference to typical question-answer characteristics in educational contexts, it appears option 4 may be a convoluted equivalent but isn’t necessarily confirmed by this analysis.
Thus, our earlier calculation stands as:
\[ (f \cdot g)(x) = \frac{5x^2 + 40x + 75}{x^2 - 9} \]
Conclusively, it seems that none of the options given are the simplified correct final answer form. Therefore, further reformulating into the context of the responses wouldn’t yield confirmation unless explicitly deciphered equivalently across rational function types. Would you like to clarify or analyze further iterations of number matches to obtain a clearer closure?
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