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. Question Help: D Post to forum

Let’s simplify the expression step by step. First, we focus on the three parts of the expression. The first part is: \[ \frac{x^{3}-125}{5 x^{2}+26 x+5} \] The numerator can be factored as a difference of cubes: \[ x^{3} - 125 = (x - 5)(x^{2} + 5x + 25) \] The denominator doesn’t factor neatly, so we'll leave it as is for now. Next, we look at: \[ \frac{x^{2}+10 x+25}{2 x^{2}+10 x+50} \] The numerator is a perfect square: \[ x^{2} + 10x + 25 = (x + 5)^{2} \] The denominator can be factored out with a 2: \[ 2x^{2} + 10x + 50 = 2(x^{2} + 5x + 25) \] Now we look at the last part: \[ \frac{x^{2}-25}{2 x-6} \] The numerator is a difference of squares: \[ x^{2} - 25 = (x - 5)(x + 5) \] Now rewriting the original expression with our factored components gives: \[ \frac{(x - 5)(x^{2} + 5x + 25)}{5x^{2} + 26x + 5} \cdot \frac{(x + 5)^{2}}{2(x^{2} + 5x + 25)} \div \frac{(x - 5)(x + 5)}{2x - 6} \] Now we can change the division to multiplication and rearrange: \[ \frac{(x - 5)(x^{2} + 5x + 25)}{5x^{2} + 26x + 5} \cdot \frac{(x + 5)^{2}}{2(x^{2} + 5x + 25)} \cdot \frac{2x - 6}{(x - 5)(x + 5)} \] Now, cancel the common factors: - \( (x - 5) \) cancels with itself. - \( (x^{2} + 5x + 25) \) cancels. - The \( 2 \) in the denominator can cancel with \( 2x - 6 = 2(x - 3) \). Ultimately, simplifying it down, we arrive at: **Numerator: \( (x + 5)(x - 3) \)** **Denominator: \( 5x^{2} + 26x + 5 \)** So, enter them as follows: Numerator: \( (x + 5)(x - 3) \) Denominator: \( 5x^{2} + 26x + 5 \)