Divide $ \dfrac{x}{x+3} $ by $ x $ to get $ \dfrac{ 1 }{ x+3 } $.

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

Step 2: Cancel $ \color{blue}{ x } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

Divide $ \dfrac{1}{x+3} $ by $ 10 $ to get $ \dfrac{ 1 }{ 10x+30 } $.

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.

$$ \begin{aligned} \frac{ \frac{1}{x+3} }{10} & \xlongequal{\text{Step 1}} \frac{1}{x+3} \cdot \frac{\color{blue}{1}}{\color{blue}{10}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot 1 }{ \left( x+3 \right) \cdot 10 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 1 }{ 10x+30 } \end{aligned} $$

Add $ \dfrac{1}{10x+30} $ and $ \dfrac{5}{x} $ to get $ \dfrac{ \color{purple}{ 51x+150 } }{ 10x^2+30x }$.

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 }$ and the second by $\color{blue}{ 10x+30 }$.

$$ \begin{aligned} \frac{1}{10x+30} + \frac{5}{x} & = \frac{ 1 \cdot \color{blue}{ x }}{ \left( 10x+30 \right) \cdot \color{blue}{ x }} + \frac{ 5 \cdot \color{blue}{ \left( 10x+30 \right) }}{ x \cdot \color{blue}{ \left( 10x+30 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x } }{ 10x^2+30x } + \frac{ \color{purple}{ 50x+150 } }{ 10x^2+30x }=\frac{ \color{purple}{ 51x+150 } }{ 10x^2+30x } \end{aligned} $$