Add $ \dfrac{3}{5} $ and $ \dfrac{x+5}{x^2} $ to get $ \dfrac{ \color{purple}{ 3x^2+5x+25 } }{ 5x^2 }$.

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^2 }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{3}{5} + \frac{x+5}{x^2} & = \frac{ 3 \cdot \color{blue}{ x^2 }}{ 5 \cdot \color{blue}{ x^2 }} + \frac{ \left( x+5 \right) \cdot \color{blue}{ 5 }}{ x^2 \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ 3x^2 } }{ 5x^2 } + \frac{ \color{purple}{ 5x+25 } }{ 5x^2 }=\frac{ \color{purple}{ 3x^2+5x+25 } }{ 5x^2 } \end{aligned} $$

Add $ \dfrac{1}{2} $ and $ \dfrac{3}{x} $ to get $ \dfrac{ \color{purple}{ x+6 } }{ 2x }$.

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 $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{1}{2} + \frac{3}{x} & = \frac{ 1 \cdot \color{blue}{ x }}{ 2 \cdot \color{blue}{ x }} + \frac{ 3 \cdot \color{blue}{ 2 }}{ x \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ x } }{ 2x } + \frac{ \color{purple}{ 6 } }{ 2x }=\frac{ \color{purple}{ x+6 } }{ 2x } \end{aligned} $$

Divide $ \dfrac{3x^2+5x+25}{5x^2} $ by $ \dfrac{x+6}{2x} $ to get $ \dfrac{ 6x^3+10x^2+50x }{ 5x^3+30x^2 } $.

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.

$$ \begin{aligned} \frac{ \frac{3x^2+5x+25}{5x^2} }{ \frac{\color{blue}{x+6}}{\color{blue}{2x}} } & \xlongequal{\text{Step 1}} \frac{3x^2+5x+25}{5x^2} \cdot \frac{\color{blue}{2x}}{\color{blue}{x+6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 3x^2+5x+25 \right) \cdot 2x }{ 5x^2 \cdot \left( x+6 \right) } \xlongequal{\text{Step 3}} \frac{ 6x^3+10x^2+50x }{ 5x^3+30x^2 } \end{aligned} $$