1. **State the problem:** Simplify the expression $$\frac{2x}{x-4} + \frac{3 \cdot 5}{2x-2}$$. 2. **Rewrite the expression:** The expression is $$\frac{2x}{x-4} + \frac{15}{2x-2}$$. 3. **Factor denominators if possible:** Note that $$2x-2 = 2(x-1)$$. 4. **Find the least common denominator (LCD):** The denominators are $$x-4$$ and $$2(x-1)$$, so the LCD is $$2(x-4)(x-1)$$. 5. **Rewrite each fraction with the LCD:** $$\frac{2x}{x-4} = \frac{2x \cdot 2(x-1)}{2(x-4)(x-1)} = \frac{4x(x-1)}{2(x-4)(x-1)}$$ $$\frac{15}{2(x-1)} = \frac{15(x-4)}{2(x-4)(x-1)}$$ 6. **Add the fractions:** $$\frac{4x(x-1)}{2(x-4)(x-1)} + \frac{15(x-4)}{2(x-4)(x-1)} = \frac{4x(x-1) + 15(x-4)}{2(x-4)(x-1)}$$ 7. **Expand the numerator:** $$4x(x-1) + 15(x-4) = 4x^2 - 4x + 15x - 60 = 4x^2 + 11x - 60$$ 8. **Factor the numerator:** Find factors of $$4 \times (-60) = -240$$ that sum to 11: 20 and -12. Rewrite: $$4x^2 + 20x - 12x - 60 = 4x(x+5) - 12(x+5) = (4x - 12)(x + 5)$$ 9. **Simplify the numerator:** $$4x - 12 = 4(x - 3)$$, so numerator is $$4(x - 3)(x + 5)$$. 10. **Write the full simplified expression:** $$\frac{4(x - 3)(x + 5)}{2(x - 4)(x - 1)}$$ 11. **Cancel common factors:** Divide numerator and denominator by 2: $$\frac{\cancel{4}(x - 3)(x + 5)}{\cancel{2}(x - 4)(x - 1)} = \frac{2(x - 3)(x + 5)}{(x - 4)(x - 1)}$$ 12. **Final answer:** $$\boxed{\frac{2(x - 3)(x + 5)}{(x - 4)(x - 1)}}$$