Multiply $ \dfrac{5}{3} $ by $ x $ to get $ \dfrac{ 5x }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{5}{3} \cdot x & \xlongequal{\text{Step 1}} \frac{5}{3} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 5 \cdot x }{ 3 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 5x }{ 3 } \end{aligned} $$

Add $ \dfrac{5x}{3} $ and $ \dfrac{x-2}{x+1} $ to get $ \dfrac{ \color{purple}{ 5x^2+8x-6 } }{ 3x+3 }$.

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 $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{5x}{3} + \frac{x-2}{x+1} & = \frac{ 5x \cdot \color{blue}{ \left( x+1 \right) }}{ 3 \cdot \color{blue}{ \left( x+1 \right) }} + \frac{ \left( x-2 \right) \cdot \color{blue}{ 3 }}{ \left( x+1 \right) \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 5x^2+5x } }{ 3x+3 } + \frac{ \color{purple}{ 3x-6 } }{ 3x+3 }=\frac{ \color{purple}{ 5x^2+8x-6 } }{ 3x+3 } \end{aligned} $$