d. \( \frac{14 x^{2}}{4 x+20} \div \frac{7 x^{2}-21 x}{x^{2}-25} \)

To simplify the expression: \[ \frac{14x^{2}}{4x + 20} \div \frac{7x^{2} - 21x}{x^{2} - 25} \] follow these steps: ### 1. Rewrite the Division as Multiplication Dividing by a fraction is the same as multiplying by its reciprocal. So, the expression becomes: \[ \frac{14x^{2}}{4x + 20} \times \frac{x^{2} - 25}{7x^{2} - 21x} \] ### 2. Factor All Possible Parts **Factor the numerator and denominator of each fraction:** - **Factor \(4x + 20\):** \[ 4x + 20 = 4(x + 5) \] - **Factor \(x^{2} - 25\) (difference of squares):** \[ x^{2} - 25 = (x - 5)(x + 5) \] - **Factor \(7x^{2} - 21x\):** \[ 7x^{2} - 21x = 7x(x - 3) \] - **Factor constants in the numerators if possible:** \[ 14x^{2} = 14x^{2} \] Now, substitute the factored forms back into the expression: \[ \frac{14x^{2}}{4(x + 5)} \times \frac{(x - 5)(x + 5)}{7x(x - 3)} \] ### 3. Simplify the Expression by Cancelling Common Factors - **Simplify the constants:** \[ \frac{14}{4} = \frac{7}{2} \] - **Cancel the \(7\) in the numerator and denominator:** - **Cancel the common factor \((x + 5)\):** The expression now looks like: \[ \frac{7x^{2}}{2} \times \frac{(x - 5)}{x(x - 3)} \] - **Simplify \(x^{2} / x = x\):** \[ \frac{7x(x - 5)}{2(x - 3)} \] ### 4. Final Simplified Expression \[ \frac{7x(x - 5)}{2(x - 3)} \] **Note:** - The simplified expression is valid provided that none of the denominators are zero. Therefore, \(x \neq -5\), \(x \neq 0\), \(x \neq 3\), and \(x \neq 5\). **Final Answer:** \[ \boxed{\dfrac{7x(x - 5)}{2(x - 3)}} \]