Subtract $ \dfrac{2x+6}{2} $ from $ \dfrac{7}{3} $ to get $ \dfrac{ \color{purple}{ -6x-4 } }{ 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}{ 2 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{7}{3} - \frac{2x+6}{2} & = \frac{ 7 \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} - \frac{ \left( 2x+6 \right) \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 14 } }{ 6 } - \frac{ \color{purple}{ 6x+18 } }{ 6 }=\frac{ \color{purple}{ 14 - \left( 6x+18 \right) } }{ 6 } = \\[1ex] &=\frac{ \color{purple}{ -6x-4 } }{ 6 } \end{aligned} $$