To find the difference between the given functions [tex]\(\frac{x}{x^2-2x-15}\)[/tex] and [tex]\(\frac{4}{x^2+2x-35}\)[/tex], we need to determine a common denominator first and then combine the fractions. Let’s simplify and analyze the given expressions step by step.
1. Factor the Denominators:
The first expression is [tex]\(\frac{x}{x^2 - 2x - 15}\)[/tex]. The denominator can be factored as:
[tex]\[
x^2 - 2x - 15 = (x - 5)(x + 3)
\][/tex]
The second expression is [tex]\(\frac{4}{x^2 + 2x - 35}\)[/tex]. The denominator can be factored as:
[tex]\[
x^2 + 2x - 35 = (x + 7)(x - 5)
\][/tex]
2. Rewrite Fractions with Common Denominator:
The common denominator for both fractions is [tex]\((x - 5)(x + 3)(x + 7)\)[/tex].
Rewrite each fraction with the common denominator:
[tex]\[
\frac{x}{(x - 5)(x + 3)} = \frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
[tex]\[
\frac{4}{(x - 5)(x + 7)} = \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
3. Combine the Fractions:
Now calculate the difference of the fractions:
[tex]\[
\frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)} - \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
Combine the numerators over the common denominator:
[tex]\[
\frac{x(x + 7) - 4(x + 3)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
4. Simplify the Numerator:
Simplify the expression in the numerator:
[tex]\[
x(x + 7) - 4(x + 3) = x^2 + 7x - 4x - 12 = x^2 + 3x - 12
\][/tex]
5. Final Combined Expression:
So the combined fraction simplifies to:
[tex]\[
\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}
\][/tex]
The final simplified difference between the given fractions is:
[tex]\[
\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}
\][/tex]
Thus, the correct answer among the options provided is:
[tex]\[
\boxed{\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}}
\][/tex]
