Решение:

1. $$\frac{x+4}{2x-6} - \frac{x+1}{x-3} = \frac{x+4}{2(x-3)} - \frac{x+1}{x-3} = \frac{x+4}{2(x-3)} - \frac{2(x+1)}{2(x-3)} = \frac{x+4 - 2(x+1)}{2(x-3)} = \frac{x+4 - 2x - 2}{2(x-3)} = \frac{-x + 2}{2(x-3)} = \frac{-(x-2)}{2(x-3)} = -\frac{x-2}{2(x-3)}$$
2. $$\frac{a+3}{3a-3} + \frac{2-a}{5a-5} = \frac{a+3}{3(a-1)} + \frac{2-a}{5(a-1)} = \frac{5(a+3)}{15(a-1)} + \frac{3(2-a)}{15(a-1)} = \frac{5a+15 + 6 - 3a}{15(a-1)} = \frac{2a + 21}{15(a-1)}$$
3. $$\frac{x+5}{x-5} - \frac{x-1}{x+5} = \frac{(x+5)(x+5)}{(x-5)(x+5)} - \frac{(x-1)(x-5)}{(x+5)(x-5)} = \frac{x^2 + 10x + 25 - (x^2 - 6x + 5)}{(x-5)(x+5)} = \frac{x^2 + 10x + 25 - x^2 + 6x - 5}{(x-5)(x+5)} = \frac{16x + 20}{(x-5)(x+5)} = \frac{4(4x + 5)}{(x-5)(x+5)}$$