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

Step 1: Write $ x^2-3x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ x }$.

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

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

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x}$.

$$ \begin{aligned} \frac{x^3-3x^2+5}{x} +2 & \xlongequal{\text{Step 1}} \frac{x^3-3x^2+5}{x} + \frac{2}{\color{red}{1}} = \frac{ x^3-3x^2+5 }{ x } + \frac{ 2 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^3-3x^2+5 } }{ x } + \frac{ \color{purple}{ 2x } }{ x }=\frac{ \color{purple}{ x^3-3x^2+2x+5 } }{ x } \end{aligned} $$