1. **State the problem:** Simplify the expression $$\frac{x + 5}{x - 4} - \frac{8x + 13}{x^2 - 3x - 4}$$. 2. **Factor the denominator:** Notice that $$x^2 - 3x - 4$$ can be factored. $$x^2 - 3x - 4 = (x - 4)(x + 1)$$. 3. **Rewrite the expression with factored denominator:** $$\frac{x + 5}{x - 4} - \frac{8x + 13}{(x - 4)(x + 1)}$$. 4. **Find common denominator:** The common denominator is $$ (x - 4)(x + 1) $$. 5. **Rewrite the first fraction with the common denominator:** $$\frac{x + 5}{x - 4} = \frac{(x + 5)(x + 1)}{(x - 4)(x + 1)}$$. 6. **Express the subtraction with common denominator:** $$\frac{(x + 5)(x + 1)}{(x - 4)(x + 1)} - \frac{8x + 13}{(x - 4)(x + 1)} = \frac{(x + 5)(x + 1) - (8x + 13)}{(x - 4)(x + 1)}$$. 7. **Expand the numerator:** $$(x + 5)(x + 1) = x^2 + x + 5x + 5 = x^2 + 6x + 5$$. 8. **Substitute and simplify numerator:** $$x^2 + 6x + 5 - (8x + 13) = x^2 + 6x + 5 - 8x - 13 = x^2 - 2x - 8$$. 9. **Factor the numerator:** $$x^2 - 2x - 8 = (x - 4)(x + 2)$$. 10. **Rewrite the expression:** $$\frac{(x - 4)(x + 2)}{(x - 4)(x + 1)}$$. 11. **Cancel common factor:** $$\frac{\cancel{(x - 4)}(x + 2)}{\cancel{(x - 4)}(x + 1)} = \frac{x + 2}{x + 1}$$. **Final answer:** $$\frac{x + 2}{x + 1}$$. **Note:** The simplification is valid for $$x \neq 4$$ and $$x \neq -1$$ because these values make the original denominators zero.