Add $ \dfrac{3x+1}{x} $ and $ \dfrac{x+1}{3} $ to get $ \dfrac{ \color{purple}{ x^2+10x+3 } }{ 3x }$.

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

$$ \begin{aligned} \frac{3x+1}{x} + \frac{x+1}{3} & = \frac{ \left( 3x+1 \right) \cdot \color{blue}{ 3 }}{ x \cdot \color{blue}{ 3 }} + \frac{ \left( x+1 \right) \cdot \color{blue}{ x }}{ 3 \cdot \color{blue}{ x }} = \\[1ex] &=\frac{ \color{purple}{ 9x+3 } }{ 3x } + \frac{ \color{purple}{ x^2+x } }{ 3x }=\frac{ \color{purple}{ x^2+10x+3 } }{ 3x } \end{aligned} $$