Multiply $ \dfrac{2}{3} $ by $ x+3 $ to get $ \dfrac{ 2x+6 }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2}{3} \cdot x+3 & \xlongequal{\text{Step 1}} \frac{2}{3} \cdot \frac{x+3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 2 \cdot \left( x+3 \right) }{ 3 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2x+6 }{ 3 } \end{aligned} $$

Multiply $ \dfrac{3}{5} $ by $ x+4 $ to get $ \dfrac{ 3x+12 }{ 5 } $.

Step 1: Write $ x+4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3}{5} \cdot x+4 & \xlongequal{\text{Step 1}} \frac{3}{5} \cdot \frac{x+4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3 \cdot \left( x+4 \right) }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3x+12 }{ 5 } \end{aligned} $$

Subtract $ \dfrac{3x+12}{5} $ from $ \dfrac{2x+6}{3} $ to get $ \dfrac{ \color{purple}{ x-6 } }{ 15 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 5 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{2x+6}{3} - \frac{3x+12}{5} & = \frac{ \left( 2x+6 \right) \cdot \color{blue}{ 5 }}{ 3 \cdot \color{blue}{ 5 }} - \frac{ \left( 3x+12 \right) \cdot \color{blue}{ 3 }}{ 5 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 10x+30 } }{ 15 } - \frac{ \color{purple}{ 9x+36 } }{ 15 }=\frac{ \color{purple}{ 10x+30 - \left( 9x+36 \right) } }{ 15 } = \\[1ex] &=\frac{ \color{purple}{ x-6 } }{ 15 } \end{aligned} $$