Simplify: \[ \frac{x^{3}-125}{5 x^{2}+26 x+5} \cdot \frac{x^{2}+10 x+25}{2 x^{2}+10 x+50} \div \frac{x^{2}-25}{2 x-6} \] Enter the numerator and denominator separately into each box below.

To simplify the expression \[ \frac{x^{3}-125}{5 x^{2}+26 x+5} \cdot \frac{x^{2}+10 x+25}{2 x^{2}+10 x+50} \div \frac{x^{2}-25}{2 x-6}, \] let's first factor each polynomial where possible: 1. **Factor \(x^{3}-125\):** This is a difference of cubes: \[ x^3 - 125 = (x - 5)(x^2 + 5x + 25). \] 2. **Factor \(x^{2}-25\):** This is a difference of squares: \[ x^2 - 25 = (x - 5)(x + 5). \] 3. **The denominator of the first fraction, \(5 x^{2}+26 x+5\), doesn't factor simply, so we can leave it as is for now.** 4. **Factor \(x^{2}+10 x+25\):** This is a perfect square: \[ x^2 + 10x + 25 = (x + 5)^2. \] 5. **Factor \(2 x^{2}+10 x+50\):** We can factor out a 2: \[ 2(x^{2}+5x+25). \] Now, let's write our expression with the factors: \[ \frac{(x - 5)(x^2 + 5x + 25)}{5 x^{2} + 26 x + 5} \cdot \frac{(x + 5)^2}{2(x^{2} + 5x + 25)} \div \frac{(x - 5)(x + 5)}{2 x - 6}. \] To facilitate the division, we can multiply by the reciprocal: \[ \frac{(x - 5)(x^2 + 5x + 25)}{5 x^{2} + 26 x + 5} \cdot \frac{(x + 5)^2}{2(x^{2} + 5x + 25)} \cdot \frac{2x - 6}{(x - 5)(x + 5)}. \] Now, let's combine all fractions into a single fraction: The expression becomes: \[ \frac{(x - 5)(x^2 + 5x + 25) \cdot (x + 5)^2 \cdot (2x - 6)}{(5 x^{2} + 26 x + 5) \cdot 2(x^{2} + 5x + 25) \cdot (x - 5)(x + 5)}. \] Next, we cancel the common factors: 1. \(x - 5\) from the numerator and denominator. 2. \(x^{2} + 5x + 25\) from the numerator and denominator. After cancellation, we get: \[ \frac{(x + 5)^2 \cdot (2x - 6)}{(5 x^{2} + 26 x + 5) \cdot 2(x + 5)}. \] Now, we can cancel one \(x + 5\): \[ \frac{(x + 5) \cdot (2x - 6)}{(5 x^{2} + 26 x + 5) \cdot 2}. \] Now in simplified terms: - **Numerator:** \((x + 5)(2x - 6)\) - **Denominator:** \(2(5 x^{2} + 26 x + 5)\) So the final answer, split into numerator and denominator, is: **Numerator:** \( (x + 5)(2x - 6) \) **Denominator:** \( 2(5 x^{2} + 26 x + 5) \)