Add $ \dfrac{5}{3x} $ and $ \dfrac{x-2}{x+1} $ to get $ \dfrac{ \color{purple}{ 3x^2-x+5 } }{ 3x^2+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}{ x+1 }$ and the second by $\color{blue}{ 3x }$.

$$ \begin{aligned} \frac{5}{3x} + \frac{x-2}{x+1} & = \frac{ 5 \cdot \color{blue}{ \left( x+1 \right) }}{ 3x \cdot \color{blue}{ \left( x+1 \right) }} + \frac{ \left( x-2 \right) \cdot \color{blue}{ 3x }}{ \left( x+1 \right) \cdot \color{blue}{ 3x }} = \\[1ex] &=\frac{ \color{purple}{ 5x+5 } }{ 3x^2+3x } + \frac{ \color{purple}{ 3x^2-6x } }{ 3x^2+3x }=\frac{ \color{purple}{ 3x^2-x+5 } }{ 3x^2+3x } \end{aligned} $$