Subtract $ \dfrac{2}{3} $ from $ \dfrac{x+2}{4} $ to get $ \dfrac{ \color{purple}{ 3x-2 } }{ 12 }$.

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 $\color{blue}{ 4 }$.

$$ \begin{aligned} \frac{x+2}{4} - \frac{2}{3} & = \frac{ \left( x+2 \right) \cdot \color{blue}{ 3 }}{ 4 \cdot \color{blue}{ 3 }} - \frac{ 2 \cdot \color{blue}{ 4 }}{ 3 \cdot \color{blue}{ 4 }} = \\[1ex] &=\frac{ \color{purple}{ 3x+6 } }{ 12 } - \frac{ \color{purple}{ 8 } }{ 12 }=\frac{ \color{purple}{ 3x-2 } }{ 12 } \end{aligned} $$

Divide $ \dfrac{3x-2}{12} $ by $ x+2 $ to get $ \dfrac{ 3x-2 }{ 12x+24 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{3x-2}{12} }{x+2} & \xlongequal{\text{Step 1}} \frac{3x-2}{12} \cdot \frac{\color{blue}{1}}{\color{blue}{x+2}} \xlongequal{\text{Step 2}} \frac{ \left( 3x-2 \right) \cdot 1 }{ 12 \cdot \left( x+2 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3x-2 }{ 12x+24 } \end{aligned} $$