Simplify. \[ \frac{2-\frac{2}{w+5}}{1-\frac{w+8}{4 w+20}} \]

To simplify the expression \[ \frac{2-\frac{2}{w+5}}{1-\frac{w+8}{4w+20}}, \] we start by simplifying both the numerator and the denominator. **Step 1: Simplify the numerator** The numerator is \[ 2 - \frac{2}{w+5}. \] To combine the terms in the numerator, we can express 2 as a fraction with a common denominator of \(w + 5\): \[ 2 = \frac{2(w+5)}{w+5}. \] So, we have: \[ 2 - \frac{2}{w+5} = \frac{2(w+5)}{w+5} - \frac{2}{w+5} = \frac{2(w+5) - 2}{w+5}. \] This simplifies to: \[ = \frac{2w + 10 - 2}{w+5} = \frac{2w + 8}{w+5}. \] **Step 2: Simplify the denominator** The denominator is \[ 1 - \frac{w+8}{4w+20}. \] Notice that \(4w + 20\) can be factored as \(4(w + 5)\). Thus, we can rewrite the denominator: \[ 1 - \frac{w + 8}{4(w + 5)}. \] Next, express 1 as a fraction: \[ 1 = \frac{4(w+5)}{4(w+5)}. \] Now combine these: \[ 1 - \frac{w+8}{4(w+5)} = \frac{4(w+5) - (w + 8)}{4(w + 5)} = \frac{4w + 20 - w - 8}{4(w + 5)} = \frac{3w + 12}{4(w + 5)}. \] **Step 3: Combine the simplified numerator and denominator** Now we can substitute our simplified forms back into the original expression: \[ \frac{\frac{2w + 8}{w + 5}}{\frac{3w + 12}{4(w + 5)}}. \] To divide by a fraction, we multiply by its reciprocal: \[ = \frac{2w + 8}{w + 5} \cdot \frac{4(w + 5)}{3w + 12}. \] The \(w + 5\) terms cancel out: \[ = \frac{4(2w + 8)}{3w + 12}. \] **Step 4: Simplify further** Notice that we can factor out a 4 from the numerator: \[ = \frac{8(w + 4)}{3(w + 4)}. \] Here, \(w + 4\) cancels out, provided \(w \neq -4\): \[ = \frac{8}{3}, \quad \text{(for } w \neq -4\text{)}. \] Thus, the simplified expression is \[ \boxed{\frac{8}{3}}. \]