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

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.

$$ \begin{aligned} \frac{ \frac{1}{3} }{x} & \xlongequal{\text{Step 1}} \frac{1}{3} \cdot \frac{\color{blue}{1}}{\color{blue}{x}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot 1 }{ 3 \cdot x } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 1 }{ 3x } \end{aligned} $$

Subtract $ \dfrac{1}{3x} $ from $ \dfrac{x^2}{75} $ to get $ \dfrac{ \color{purple}{ 3x^3-75 } }{ 225x }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 3x }$ and the second by $\color{blue}{ 75 }$.

$$ \begin{aligned} \frac{x^2}{75} - \frac{1}{3x} & = \frac{ x^2 \cdot \color{blue}{ 3x }}{ 75 \cdot \color{blue}{ 3x }} - \frac{ 1 \cdot \color{blue}{ 75 }}{ 3x \cdot \color{blue}{ 75 }} = \\[1ex] &=\frac{ \color{purple}{ 3x^3 } }{ 225x } - \frac{ \color{purple}{ 75 } }{ 225x }=\frac{ \color{purple}{ 3x^3-75 } }{ 225x } \end{aligned} $$

Add $ \dfrac{3x^3-75}{225x} $ and $ \dfrac{5}{12} $ to get $ \dfrac{ \color{purple}{ 36x^3+1125x-900 } }{ 2700x }$.

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}{ 12 }$ and the second by $\color{blue}{ 225x }$.

$$ \begin{aligned} \frac{3x^3-75}{225x} + \frac{5}{12} & = \frac{ \left( 3x^3-75 \right) \cdot \color{blue}{ 12 }}{ 225x \cdot \color{blue}{ 12 }} + \frac{ 5 \cdot \color{blue}{ 225x }}{ 12 \cdot \color{blue}{ 225x }} = \\[1ex] &=\frac{ \color{purple}{ 36x^3-900 } }{ 2700x } + \frac{ \color{purple}{ 1125x } }{ 2700x }=\frac{ \color{purple}{ 36x^3+1125x-900 } }{ 2700x } \end{aligned} $$