1. **State the problem:** Simplify the expression $$\frac{1 - \frac{4}{x+8}}{x + \frac{16}{x+8}}$$. 2. **Rewrite the expression:** To simplify, write the numerator and denominator with a common denominator: Numerator: $$1 - \frac{4}{x+8} = \frac{x+8}{x+8} - \frac{4}{x+8} = \frac{x+8-4}{x+8} = \frac{x+4}{x+8}$$ Denominator: $$x + \frac{16}{x+8} = \frac{x(x+8)}{x+8} + \frac{16}{x+8} = \frac{x^2 + 8x + 16}{x+8}$$ 3. **Rewrite the entire expression:** $$\frac{\frac{x+4}{x+8}}{\frac{x^2 + 8x + 16}{x+8}}$$ 4. **Divide the fractions:** $$= \frac{x+4}{x+8} \times \frac{x+8}{x^2 + 8x + 16}$$ 5. **Cancel common factors:** The factor $x+8$ appears in numerator and denominator: $$= \frac{x+4}{\cancel{x+8}} \times \frac{\cancel{x+8}}{x^2 + 8x + 16} = \frac{x+4}{x^2 + 8x + 16}$$ 6. **Factor the denominator:** $$x^2 + 8x + 16 = (x+4)^2$$ 7. **Simplify the fraction:** $$\frac{x+4}{(x+4)^2} = \frac{\cancel{x+4}}{(\cancel{x+4})(x+4)} = \frac{1}{x+4}$$ **Final answer:** $$\boxed{\frac{1}{x+4}}$$