Answer
$ \frac{x+3}{x+5}$.
Work Step by Step
First determine the Least Common denominator (LCD).
Factor $x^2-25$.
$\Rightarrow x^2-5^2$
Use special formula $A^2-B^2=(A+B)(A-B)$.
$\Rightarrow (x+5)(x-5)$
Back substitute into the given expression.
$\Rightarrow \frac{x-4}{x-5}-\frac{3}{x+5}-\frac{10}{x^2-25}=\frac{x-4}{x-5}-\frac{3}{x+5}-\frac{10}{(x+5)(x-5)}$
The LCD is $(x+5)(x-5)$.
Multiply the numerator and the denominator to form LCD at the denominators.
$\Rightarrow \frac{(x-4)(x+5)}{(x+5)(x-5)}-\frac{3(x-5)}{(x+5)(x-5)}-\frac{10}{(x+5)(x-5)}$
Use the FOIL method and the distributive property.
$\Rightarrow \frac{x^2+5x-4x-20}{(x+5)(x-5)}-\frac{3x-15}{(x+5)(x-5)}-\frac{10}{(x+5)(x-5)}$
Add numerators because denominators are the same.
$\Rightarrow \frac{x^2+5x-4x-20-(3x-15)-10}{(x+5)(x-5)}$
Simplify.
$\Rightarrow \frac{x^2+5x-4x-20-3x+15-10}{(x+5)(x-5)}$
Add like terms.
$\Rightarrow \frac{x^2-2x-15}{(x+5)(x-5)}$
Factor $x^2-2x-15$.
Rewrite the middle term $-2x$ as $-5x+3x$.
$\Rightarrow x^2-5x+3x-15$
Group terms.
$\Rightarrow (x^2-5x)+(3x-15)$
Factor each group.
$\Rightarrow x(x-5)+3(x-5)$
Factor out $(x-5)$.
$\Rightarrow (x-5)(x+3)$
Back substitute the factor into the fraction.
$\Rightarrow \frac{(x-5)(x+3)}{(x+5)(x-5)}$
Cancel common factors.
$\Rightarrow \frac{x+3}{x+5}$.
