Simplify. \[ \frac{1-\frac{4}{x+8}}{\frac{16}{x+8}+x} \]

To simplify the expression \[ \frac{1-\frac{4}{x+8}}{\frac{16}{x+8}+x}, \] we will work on both the numerator and the denominator separately. **Step 1: Simplify the numerator.** The numerator is \( 1 - \frac{4}{x+8} \). To combine these terms, we need a common denominator, which is \( x + 8 \). \[ 1 = \frac{x+8}{x+8} \] Thus, we have: \[ 1 - \frac{4}{x+8} = \frac{x+8}{x+8} - \frac{4}{x+8} = \frac{x + 8 - 4}{x + 8} = \frac{x + 4}{x + 8}. \] **Step 2: Simplify the denominator.** The denominator is \( \frac{16}{x+8} + x \). Again, we need a common denominator. \[ x = \frac{x(x + 8)}{x + 8} = \frac{x^2 + 8x}{x + 8}. \] Now we can add: \[ \frac{16}{x + 8} + x = \frac{16}{x + 8} + \frac{x^2 + 8x}{x + 8} = \frac{16 + x^2 + 8x}{x + 8} = \frac{x^2 + 8x + 16}{x + 8}. \] The expression \( x^2 + 8x + 16 \) can be factored using the perfect square formula: \[ x^2 + 8x + 16 = (x + 4)^2. \] So the denominator becomes: \[ \frac{(x + 4)^2}{x + 8}. \] **Step 3: Combine the simplified numerator and denominator.** Now we can put everything together: \[ \frac{\frac{x + 4}{x + 8}}{\frac{(x + 4)^2}{x + 8}}. \] When we divide by a fraction, we multiply by its reciprocal: \[ = \frac{x + 4}{x + 8} \cdot \frac{x + 8}{(x + 4)^2}. \] The \( x + 8 \) cancels out: \[ = \frac{x + 4}{(x + 4)^2} = \frac{1}{x + 4}, \quad (x + 4 \neq 0). \] Thus, the simplified expression is: \[ \boxed{\frac{1}{x + 4}}. \]