1. **State the problem:** Solve the equation $$\frac{8}{x} + \frac{1}{x+2} = -2$$ for $x$. 2. **Find a common denominator:** The denominators are $x$ and $x+2$. The common denominator is $x(x+2)$. 3. **Rewrite each term with the common denominator:** $$\frac{8(x+2)}{x(x+2)} + \frac{1 \cdot x}{x(x+2)} = -2$$ 4. **Combine the fractions:** $$\frac{8(x+2) + x}{x(x+2)} = -2$$ 5. **Multiply both sides by $x(x+2)$ to clear the denominator:** $$8(x+2) + x = -2x(x+2)$$ 6. **Expand both sides:** $$8x + 16 + x = -2x^2 - 4x$$ 7. **Combine like terms on the left:** $$9x + 16 = -2x^2 - 4x$$ 8. **Bring all terms to one side to set equation to zero:** $$2x^2 + 9x + 4x + 16 = 0$$ $$2x^2 + 13x + 16 = 0$$ 9. **Use the quadratic formula to solve for $x$:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=13$, and $c=16$. 10. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 13^2 - 4 \cdot 2 \cdot 16 = 169 - 128 = 41$$ 11. **Find the roots:** $$x = \frac{-13 \pm \sqrt{41}}{2 \cdot 2} = \frac{-13 \pm \sqrt{41}}{4}$$ 12. **Final answer:** $$x = \frac{-13 + \sqrt{41}}{4}, \frac{-13 - \sqrt{41}}{4}$$ These are the simplified exact solutions.