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

Step 1: Write $ x $ 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 $\color{blue}{ x }$.

$$ \begin{aligned} x+ \frac{4}{x} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} + \frac{4}{x} = \frac{ x \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 4 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2 } }{ x } + \frac{ \color{purple}{ 4 } }{ x }=\frac{ \color{purple}{ x^2+4 } }{ x } \end{aligned} $$

Subtract $3$ from $ \dfrac{x^2+4}{x} $ to get $ \dfrac{ \color{purple}{ x^2-3x+4 } }{ x }$.

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

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

$$ \begin{aligned} \frac{x^2+4}{x} -3 & \xlongequal{\text{Step 1}} \frac{x^2+4}{x} - \frac{3}{\color{red}{1}} = \frac{ x^2+4 }{ x } - \frac{ 3 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2+4 } }{ x } - \frac{ \color{purple}{ 3x } }{ x }=\frac{ \color{purple}{ x^2-3x+4 } }{ x } \end{aligned} $$