Subtract $ \dfrac{x+4}{3} $ from $ \dfrac{x-3}{2} $ to get $ \dfrac{ \color{purple}{ x-17 } }{ 6 }$.

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}{ 3 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{x-3}{2} - \frac{x+4}{3} & = \frac{ \left( x-3 \right) \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} - \frac{ \left( x+4 \right) \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3x-9 } }{ 6 } - \frac{ \color{purple}{ 2x+8 } }{ 6 }=\frac{ \color{purple}{ 3x-9 - \left( 2x+8 \right) } }{ 6 } = \\[1ex] &=\frac{ \color{purple}{ x-17 } }{ 6 } \end{aligned} $$