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

Step 1: Write $ 3a $ 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 $\color{blue}{ a }$.

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

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

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 $\color{blue}{a}$.

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