1. **State the problem:** Add the rational expressions $$\frac{5}{x-7} + \frac{x-3}{x-4}$$ and simplify the result as much as possible. 2. **Formula and rules:** To add rational expressions, find a common denominator, which is usually the least common denominator (LCD) of the denominators. 3. **Find the LCD:** The denominators are $x-7$ and $x-4$. The LCD is $(x-7)(x-4)$. 4. **Rewrite each fraction with the LCD:** $$\frac{5}{x-7} = \frac{5(x-4)}{(x-7)(x-4)}$$ $$\frac{x-3}{x-4} = \frac{(x-3)(x-7)}{(x-4)(x-7)}$$ 5. **Add the numerators over the common denominator:** $$\frac{5(x-4) + (x-3)(x-7)}{(x-7)(x-4)}$$ 6. **Expand the numerators:** $$5(x-4) = 5x - 20$$ $$(x-3)(x-7) = x^2 - 7x - 3x + 21 = x^2 - 10x + 21$$ 7. **Combine the expanded numerators:** $$5x - 20 + x^2 - 10x + 21 = x^2 - 5x + 1$$ 8. **Write the final expression:** $$\frac{x^2 - 5x + 1}{(x-7)(x-4)}$$ 9. **Check for further simplification:** The numerator $x^2 - 5x + 1$ does not factor nicely, so this is the simplified form. **Final answer:** $$\frac{x^2 - 5x + 1}{(x-7)(x-4)}$$