Simplify $ \dfrac{2x^2+10x}{x^2+6x+5} $ to $ \dfrac{2x}{x+1} $.

Factor both the denominator and the numerator, then cancel the common factor. $\color{blue}{x+5}$.

$$ \begin{aligned} \frac{2x^2+10x}{x^2+6x+5} & =\frac{ 2x \cdot \color{blue}{ \left( x+5 \right) }}{ \left( x+1 \right) \cdot \color{blue}{ \left( x+5 \right) }} = \\[1ex] &= \frac{2x}{x+1} \end{aligned} $$

Divide $ \dfrac{2x}{x+1} $ by $ \dfrac{x^2+6x+9}{x^2-x-2} $ to get $ \dfrac{ 2x^2-4x }{ x^2+6x+9 } $.

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.

$$ \begin{aligned} \frac{ \frac{2x}{x+1} }{ \frac{\color{blue}{x^2+6x+9}}{\color{blue}{x^2-x-2}} } & \xlongequal{\text{Step 1}} \frac{2x}{x+1} \cdot \frac{\color{blue}{x^2-x-2}}{\color{blue}{x^2+6x+9}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 2x }{ 1 \cdot \color{red}{ \left( x+1 \right) } } \cdot \frac{ \left( x-2 \right) \cdot \color{red}{ \left( x+1 \right) } }{ x^2+6x+9 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2x }{ 1 } \cdot \frac{ x-2 }{ x^2+6x+9 } \xlongequal{\text{Step 4}} \frac{ 2x \cdot \left( x-2 \right) }{ 1 \cdot \left( x^2+6x+9 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ 2x^2-4x }{ x^2+6x+9 } \end{aligned} $$