Multiply $3$ by $ \dfrac{d}{d+8} $ to get $ \dfrac{ 3d }{ d+8 } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{d}{d+8} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{d}{d+8} \xlongequal{\text{Step 2}} \frac{ 3 \cdot d }{ 1 \cdot \left( d+8 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3d }{ d+8 } \end{aligned} $$

Add $ \dfrac{3}{d+8} $ and $ \dfrac{3}{d-8} $ to get $ \dfrac{ \color{purple}{ 6d } }{ d^2-64 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ d-8 }$ and the second by $\color{blue}{ d+8 }$.

$$ \begin{aligned} \frac{3}{d+8} + \frac{3}{d-8} & = \frac{ 3 \cdot \color{blue}{ \left( d-8 \right) }}{ \left( d+8 \right) \cdot \color{blue}{ \left( d-8 \right) }} + \frac{ 3 \cdot \color{blue}{ \left( d+8 \right) }}{ \left( d-8 \right) \cdot \color{blue}{ \left( d+8 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3d-24 } }{ d^2 -\cancel{8d}+ \cancel{8d}-64 } + \frac{ \color{purple}{ 3d+24 } }{ d^2 -\cancel{8d}+ \cancel{8d}-64 } = \\[1ex] &=\frac{ \color{purple}{ 6d } }{ d^2-64 } \end{aligned} $$

Subtract $ \dfrac{3d}{d+8} $ from $ d $ to get $ \dfrac{ \color{purple}{ d^2+5d } }{ d+8 }$.

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

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ d+8 }$.

$$ \begin{aligned} d- \frac{3d}{d+8} & \xlongequal{\text{Step 1}} \frac{d}{\color{red}{1}} - \frac{3d}{d+8} = \frac{ d \cdot \color{blue}{ \left( d+8 \right) }}{ 1 \cdot \color{blue}{ \left( d+8 \right) }} - \frac{ 3d }{ d+8 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ d^2+8d } }{ d+8 } - \frac{ \color{purple}{ 3d } }{ d+8 }=\frac{ \color{purple}{ d^2+5d } }{ d+8 } \end{aligned} $$

Add $ \dfrac{3}{d+8} $ and $ \dfrac{3}{d-8} $ to get $ \dfrac{ \color{purple}{ 6d } }{ d^2-64 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ d-8 }$ and the second by $\color{blue}{ d+8 }$.

$$ \begin{aligned} \frac{3}{d+8} + \frac{3}{d-8} & = \frac{ 3 \cdot \color{blue}{ \left( d-8 \right) }}{ \left( d+8 \right) \cdot \color{blue}{ \left( d-8 \right) }} + \frac{ 3 \cdot \color{blue}{ \left( d+8 \right) }}{ \left( d-8 \right) \cdot \color{blue}{ \left( d+8 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3d-24 } }{ d^2 -\cancel{8d}+ \cancel{8d}-64 } + \frac{ \color{purple}{ 3d+24 } }{ d^2 -\cancel{8d}+ \cancel{8d}-64 } = \\[1ex] &=\frac{ \color{purple}{ 6d } }{ d^2-64 } \end{aligned} $$

Divide $ \dfrac{d^2+5d}{d+8} $ by $ \dfrac{6d}{d^2-64} $ to get $ \dfrac{d^3-3d^2-40d}{6d} $.

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

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{d^2+5d}{d+8} }{ \frac{\color{blue}{6d}}{\color{blue}{d^2-64}} } & \xlongequal{\text{Step 1}} \frac{d^2+5d}{d+8} \cdot \frac{\color{blue}{d^2-64}}{\color{blue}{6d}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ d^2+5d }{ 1 \cdot \color{red}{ \left( d+8 \right) } } \cdot \frac{ \left( d-8 \right) \cdot \color{red}{ \left( d+8 \right) } }{ 6d } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ d^2+5d }{ 1 } \cdot \frac{ d-8 }{ 6d } \xlongequal{\text{Step 4}} \frac{ \left( d^2+5d \right) \cdot \left( d-8 \right) }{ 1 \cdot 6d } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ d^3-8d^2+5d^2-40d }{ 6d } = \frac{d^3-3d^2-40d}{6d} \end{aligned} $$