#### Answer

$$\frac{-2x-14}{\left(x+2\right)\left(x-4\right)\left(x-8\right)}$$

#### Work Step by Step

Simplifying the expression by creating like denominators, we find:
$$\frac{x+3}{\left(x+2\right)\left(x-4\right)}-\frac{x-5}{\left(x-4\right)\left(x-8\right)} \\ \frac{\left(x+3\right)\left(x-8\right)}{\left(x+2\right)\left(x-4\right)\left(x-8\right)}-\frac{\left(x-5\right)\left(x+2\right)}{\left(x-4\right)\left(x-8\right)\left(x+2\right)} \\ \frac{\left(x+3\right)\left(x-8\right)-\left(x-5\right)\left(x+2\right)}{\left(x+2\right)\left(x-4\right)\left(x-8\right)}\\ \frac{-2x-14}{\left(x+2\right)\left(x-4\right)\left(x-8\right)}$$