Divide $ \dfrac{2x+4}{x-5} $ by $ 4x+20 $ to get $ \dfrac{x+2}{2x^2-50} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{2x+4}{x-5} }{4x+20} & \xlongequal{\text{Step 1}} \frac{2x+4}{x-5} \cdot \frac{\color{blue}{1}}{\color{blue}{4x+20}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x+2 \right) \cdot \color{blue}{2} }{ x-5 } \cdot \frac{ 1 }{ \left( 2x+10 \right) \cdot \color{blue}{2} } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x+2 }{ x-5 } \cdot \frac{ 1 }{ 2x+10 } \xlongequal{\text{Step 4}} \frac{ \left( x+2 \right) \cdot 1 }{ \left( x-5 \right) \cdot \left( 2x+10 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ x+2 }{ 2x^2+ \cancel{10x} -\cancel{10x}-50 } = \frac{x+2}{2x^2-50} \end{aligned} $$

Divide $ \dfrac{x+2}{2x^2-50} $ by $ x^2-25 $ to get $ \dfrac{x+2}{2x^4-100x^2+1250} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{x+2}{2x^2-50} }{x^2-25} & \xlongequal{\text{Step 1}} \frac{x+2}{2x^2-50} \cdot \frac{\color{blue}{1}}{\color{blue}{x^2-25}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x+2 \right) \cdot 1 }{ \left( 2x^2-50 \right) \cdot \left( x^2-25 \right) } \xlongequal{\text{Step 3}} \frac{ x+2 }{ 2x^4-50x^2-50x^2+1250 } = \\[1ex] &= \frac{x+2}{2x^4-100x^2+1250} \end{aligned} $$