Subtract $ \dfrac{4}{x+2} $ from $ \dfrac{1}{3x} $ to get $ \dfrac{ \color{purple}{ -11x+2 } }{ 3x^2+6x }$.

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

$$ \begin{aligned} \frac{1}{3x} - \frac{4}{x+2} & = \frac{ 1 \cdot \color{blue}{ \left( x+2 \right) }}{ 3x \cdot \color{blue}{ \left( x+2 \right) }} - \frac{ 4 \cdot \color{blue}{ 3x }}{ \left( x+2 \right) \cdot \color{blue}{ 3x }} = \\[1ex] &=\frac{ \color{purple}{ x+2 } }{ 3x^2+6x } - \frac{ \color{purple}{ 12x } }{ 3x^2+6x }=\frac{ \color{purple}{ -11x+2 } }{ 3x^2+6x } \end{aligned} $$