1. **State the problem:** Simplify the expression $$\frac{4 + \frac{8}{x^5}}{\frac{x^2}{9} - \frac{4}{x}} \times \left(\frac{\frac{x}{2} + \frac{2}{x} - 1}{\frac{x^2}{4} + \frac{x}{2} + 1}\right) \times \frac{1 + \frac{2}{x}}{x - 2}$$ 2. **Rewrite all terms with common denominators:** - Numerator of first fraction: $4 + \frac{8}{x^5} = \frac{4x^5}{x^5} + \frac{8}{x^5} = \frac{4x^5 + 8}{x^5}$ - Denominator of first fraction: $\frac{x^2}{9} - \frac{4}{x} = \frac{x^3}{9x} - \frac{36}{9x} = \frac{x^3 - 36}{9x}$ So first fraction becomes: $$\frac{\frac{4x^5 + 8}{x^5}}{\frac{x^3 - 36}{9x}} = \frac{4x^5 + 8}{x^5} \times \frac{9x}{x^3 - 36} = \frac{9x(4x^5 + 8)}{x^5(x^3 - 36)}$$ 3. **Simplify numerator $4x^5 + 8$ by factoring out 4:** $$4x^5 + 8 = 4(x^5 + 2)$$ 4. **Simplify denominator $x^3 - 36$ if possible:** No simple factorization over integers, so keep as is. 5. **Simplify the second fraction:** Numerator: $\frac{x}{2} + \frac{2}{x} - 1 = \frac{x^2}{2x} + \frac{4}{2x} - \frac{2x}{2x} = \frac{x^2 + 4 - 2x}{2x} = \frac{x^2 - 2x + 4}{2x}$ Denominator: $\frac{x^2}{4} + \frac{x}{2} + 1 = \frac{x^2}{4} + \frac{2x}{4} + \frac{4}{4} = \frac{x^2 + 2x + 4}{4}$ So second fraction is: $$\frac{\frac{x^2 - 2x + 4}{2x}}{\frac{x^2 + 2x + 4}{4}} = \frac{x^2 - 2x + 4}{2x} \times \frac{4}{x^2 + 2x + 4} = \frac{4(x^2 - 2x + 4)}{2x(x^2 + 2x + 4)} = \frac{2(x^2 - 2x + 4)}{x(x^2 + 2x + 4)}$$ 6. **Simplify the third fraction:** $$\frac{1 + \frac{2}{x}}{x - 2} = \frac{\frac{x + 2}{x}}{x - 2} = \frac{x + 2}{x(x - 2)}$$ 7. **Combine all three parts:** $$\frac{9x \cdot 4(x^5 + 2)}{x^5 (x^3 - 36)} \times \frac{2(x^2 - 2x + 4)}{x(x^2 + 2x + 4)} \times \frac{x + 2}{x(x - 2)} = \frac{36x (x^5 + 2)}{x^5 (x^3 - 36)} \times \frac{2(x^2 - 2x + 4)}{x(x^2 + 2x + 4)} \times \frac{x + 2}{x(x - 2)}$$ Multiply numerators and denominators: Numerator: $$36x (x^5 + 2) \times 2(x^2 - 2x + 4) \times (x + 2) = 72x (x^5 + 2)(x^2 - 2x + 4)(x + 2)$$ Denominator: $$x^5 (x^3 - 36) \times x (x^2 + 2x + 4) \times x (x - 2) = x^7 (x^3 - 36)(x^2 + 2x + 4)(x - 2)$$ 8. **Final simplified expression:** $$\frac{72x (x^5 + 2)(x^2 - 2x + 4)(x + 2)}{x^7 (x^3 - 36)(x^2 + 2x + 4)(x - 2)} = \frac{72 (x^5 + 2)(x^2 - 2x + 4)(x + 2)}{x^6 (x^3 - 36)(x^2 + 2x + 4)(x - 2)}$$ This is the simplified form, with all factors shown. **Note:** No further factorization is straightforward. **Answer:** $$\boxed{\frac{72 (x^5 + 2)(x^2 - 2x + 4)(x + 2)}{x^6 (x^3 - 36)(x^2 + 2x + 4)(x - 2)}}$$