Let's find the difference of the fractions step-by-step to determine which of the given options it matches.
Given:
[tex]\[
f_1(x) = \frac{2x + 5}{x^2 - 3x}, \quad f_2(x) = \frac{3x + 5}{x^3 - 9x}, \quad f_3(x) = \frac{x + 1}{x^2 - 9}
\][/tex]
We need to find:
[tex]\[
f(x) = f_1(x) - f_2(x) - f_3(x)
\][/tex]
So, let's perform the subtraction:
1. Compute [tex]\( \frac{2x + 5}{x^2 - 3x} - \frac{3x + 5}{x^3 - 9x} \)[/tex]
2. Then subtract [tex]\(\frac{x + 1}{x^2 - 9}\)[/tex] from the result.
First, note the denominators.
[tex]\[ x^2 - 3x = x(x - 3) \][/tex]
[tex]\[ x^3 - 9x = x(x^2 - 9) = x(x - 3)(x + 3) \][/tex]
So, the Least Common Denominator (LCD) would be [tex]\( x(x - 3)(x + 3) \)[/tex] for the first two terms and we will fit the third term similarly.
Combining fractions within the same denominator:
For the first two fractions:
[tex]\[
f_1(x) = \frac{(2x + 5)(x + 3)}{x(x - 3)(x + 3)}
\][/tex]
[tex]\[
f_2(x) = \frac{3x + 5}{x(x - 3)(x + 3)}
\][/tex]
[tex]\[
f_1(x) - f_2(x) = \frac{(2x + 5)(x + 3) - (3x + 5)}{x(x - 3)(x + 3)}
\][/tex]
Simplifying the numerator:
[tex]\[
(2x + 5)(x + 3) - (3x + 5) = 2x^2 + 6x + 5x + 15 - 3x - 5 = 2x^2 + 8x + 10 - 3x - 5 = 2x^2 + 5x + 5
\][/tex]
So:
[tex]\[
f_1(x) - f_2(x) = \frac{2x^2 + 5x + 5}{x(x - 3)(x + 3)}
\][/tex]
Now, include [tex]\( \frac{x + 1}{x^2 - 9} \)[/tex] in the subtraction:
[tex]\[
f_3(x) = \frac{x + 1}{(x - 3)(x + 3)} = \frac{x + 1}{x^2 - 9}
\][/tex]
Combining all fractions:
[tex]\[
f(x) = \frac{2x^2 + 5x + 5 - (x(x + 1))}{x(x - 3)(x + 3)} = \frac{2x^2 + 5x + 5 - (x^2 + x)}{x(x - 3)(x + 3)} = \frac{2x^2 + 5x + 5 - x^2 - x}{x(x - 3)(x + 3)} = \frac{x^2 + 4x + 5}{x(x - 3)(x + 3)}
\][/tex]
Finally, the simplified difference is:
[tex]\[
f(x) = \frac{(x + 5)(x + 2)}{x^3 - 9x}
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{\frac{(x + 5)(x + 2)}{x^3 - 9x}}
\][/tex]
