Use the distributive property.

Multiply $ \color{blue}{4} $ by $ \left( x+2\right) $ $$ \color{blue}{4} \cdot \left( x+2\right) = 4x+8 $$

Subtract $ \dfrac{2}{3} $ from $ \dfrac{1}{4x+8} $ to get $ \dfrac{ \color{purple}{ -8x-13 } }{ 12x+24 }$.

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 $\color{blue}{ 4x+8 }$.

$$ \begin{aligned} \frac{1}{4x+8} - \frac{2}{3} & = \frac{ 1 \cdot \color{blue}{ 3 }}{ \left( 4x+8 \right) \cdot \color{blue}{ 3 }} - \frac{ 2 \cdot \color{blue}{ \left( 4x+8 \right) }}{ 3 \cdot \color{blue}{ \left( 4x+8 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3 } }{ 12x+24 } - \frac{ \color{purple}{ 8x+16 } }{ 12x+24 }=\frac{ \color{purple}{ 3 - \left( 8x+16 \right) } }{ 12x+24 } = \\[1ex] &=\frac{ \color{purple}{ -8x-13 } }{ 12x+24 } \end{aligned} $$

Multiply $ \dfrac{-8x-13}{12x+24} $ by $ x+2 $ to get $ \dfrac{ -8x-13 }{ 12 } $.

Step 1: Write $ x+2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{-8x-13}{12x+24} \cdot x+2 & \xlongequal{\text{Step 1}} \frac{-8x-13}{12x+24} \cdot \frac{x+2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ -8x-13 }{ 12 \cdot \color{red}{ \left( x+2 \right) } } \cdot \frac{ 1 \cdot \color{red}{ \left( x+2 \right) } }{ 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -8x-13 }{ 12 } \cdot \frac{ 1 }{ 1 } \xlongequal{\text{Step 4}} \frac{ \left( -8x-13 \right) \cdot 1 }{ 12 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ -8x-13 }{ 12 } \end{aligned} $$