Add $ \dfrac{4}{x} $ and $ \dfrac{x}{x+1} $ to get $ \dfrac{ \color{purple}{ x^2+4x+4 } }{ x^2+x }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ x+1 }$ and the second by $\color{blue}{ x }$.

$$ \begin{aligned} \frac{4}{x} + \frac{x}{x+1} & = \frac{ 4 \cdot \color{blue}{ \left( x+1 \right) }}{ x \cdot \color{blue}{ \left( x+1 \right) }} + \frac{ x \cdot \color{blue}{ x }}{ \left( x+1 \right) \cdot \color{blue}{ x }} = \\[1ex] &=\frac{ \color{purple}{ 4x+4 } }{ x^2+x } + \frac{ \color{purple}{ x^2 } }{ x^2+x }=\frac{ \color{purple}{ x^2+4x+4 } }{ x^2+x } \end{aligned} $$

Divide $ \dfrac{x^2+4x+4}{x^2+x} $ by $ \dfrac{x^2-3x-10}{5x} $ to get $ \dfrac{5x^2+10x}{x^3-4x^2-5x} $.

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.

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