1. **State the problem:** Simplify the expression $$\frac{X+\frac{2x}{x-2}}{1+\frac{4}{x^2}-4}$$. 2. **Rewrite the expression:** The numerator is $$X + \frac{2x}{x-2}$$ and the denominator is $$1 + \frac{4}{x^2} - 4$$. 3. **Simplify the denominator:** Combine like terms: $$1 - 4 + \frac{4}{x^2} = -3 + \frac{4}{x^2}$$ 4. **Rewrite the denominator with a common denominator:** $$-3 + \frac{4}{x^2} = \frac{-3x^2}{x^2} + \frac{4}{x^2} = \frac{-3x^2 + 4}{x^2}$$ 5. **Rewrite the entire expression:** $$\frac{X + \frac{2x}{x-2}}{\frac{-3x^2 + 4}{x^2}} = \left(X + \frac{2x}{x-2}\right) \times \frac{x^2}{-3x^2 + 4}$$ 6. **Simplify the numerator:** Find common denominator for $$X$$ and $$\frac{2x}{x-2}$$: $$X = \frac{X(x-2)}{x-2}$$ So, $$X + \frac{2x}{x-2} = \frac{X(x-2) + 2x}{x-2} = \frac{Xx - 2X + 2x}{x-2}$$ 7. **Rewrite the expression:** $$\frac{Xx - 2X + 2x}{x-2} \times \frac{x^2}{-3x^2 + 4} = \frac{(Xx - 2X + 2x) x^2}{(x-2)(-3x^2 + 4)}$$ 8. **Factor denominator term:** Note that $$-3x^2 + 4 = -(3x^2 - 4)$$ and $$3x^2 - 4$$ does not factor nicely over integers. 9. **Final simplified form:** $$\frac{(Xx - 2X + 2x) x^2}{(x-2)(-3x^2 + 4)}$$ This is the simplified expression. **Answer:** $$\boxed{\frac{(Xx - 2X + 2x) x^2}{(x-2)(-3x^2 + 4)}}$$