1. **State the problem:** Simplify the algebraic expression $$\frac{2}{x-2} + \frac{3}{x^3 - 4x}$$. 2. **Factor the denominator:** Notice that $$x^3 - 4x$$ can be factored by taking out the common factor $$x$$: $$x^3 - 4x = x(x^2 - 4)$$. 3. **Further factor the quadratic:** Recognize $$x^2 - 4$$ as a difference of squares: $$x^2 - 4 = (x-2)(x+2)$$. 4. **Rewrite the expression:** Substitute the factored form back: $$\frac{2}{x-2} + \frac{3}{x(x-2)(x+2)}$$. 5. **Find the common denominator:** The least common denominator (LCD) is $$x(x-2)(x+2)$$. 6. **Rewrite each fraction with the LCD:** $$\frac{2}{x-2} = \frac{2 \cdot x(x+2)}{x(x-2)(x+2)} = \frac{2x(x+2)}{x(x-2)(x+2)}$$ 7. **Add the fractions:** $$\frac{2x(x+2)}{x(x-2)(x+2)} + \frac{3}{x(x-2)(x+2)} = \frac{2x(x+2) + 3}{x(x-2)(x+2)}$$ 8. **Expand the numerator:** $$2x(x+2) + 3 = 2x^2 + 4x + 3$$ 9. **Check if numerator can be factored:** The quadratic $$2x^2 + 4x + 3$$ does not factor nicely with integer factors. 10. **Final simplified expression:** $$\boxed{\frac{2x^2 + 4x + 3}{x(x-2)(x+2)}}$$