Multiply $4$ by $ \dfrac{w}{w+7} $ to get $ \dfrac{ 4w }{ w+7 } $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{w}{w+7} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{w}{w+7} \xlongequal{\text{Step 2}} \frac{ 4 \cdot w }{ 1 \cdot \left( w+7 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4w }{ w+7 } \end{aligned} $$

Add $ \dfrac{4}{w+7} $ and $ \dfrac{4}{w-7} $ to get $ \dfrac{ \color{purple}{ 8w } }{ w^2-49 }$.

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}{ w-7 }$ and the second by $\color{blue}{ w+7 }$.

$$ \begin{aligned} \frac{4}{w+7} + \frac{4}{w-7} & = \frac{ 4 \cdot \color{blue}{ \left( w-7 \right) }}{ \left( w+7 \right) \cdot \color{blue}{ \left( w-7 \right) }} + \frac{ 4 \cdot \color{blue}{ \left( w+7 \right) }}{ \left( w-7 \right) \cdot \color{blue}{ \left( w+7 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 4w-28 } }{ w^2 -\cancel{7w}+ \cancel{7w}-49 } + \frac{ \color{purple}{ 4w+28 } }{ w^2 -\cancel{7w}+ \cancel{7w}-49 } = \\[1ex] &=\frac{ \color{purple}{ 8w } }{ w^2-49 } \end{aligned} $$

Subtract $ \dfrac{4w}{w+7} $ from $ w $ to get $ \dfrac{ \color{purple}{ w^2+3w } }{ w+7 }$.

Step 1: Write $ w $ 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 $\color{blue}{ w+7 }$.

$$ \begin{aligned} w- \frac{4w}{w+7} & \xlongequal{\text{Step 1}} \frac{w}{\color{red}{1}} - \frac{4w}{w+7} = \frac{ w \cdot \color{blue}{ \left( w+7 \right) }}{ 1 \cdot \color{blue}{ \left( w+7 \right) }} - \frac{ 4w }{ w+7 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ w^2+7w } }{ w+7 } - \frac{ \color{purple}{ 4w } }{ w+7 }=\frac{ \color{purple}{ w^2+3w } }{ w+7 } \end{aligned} $$

Add $ \dfrac{4}{w+7} $ and $ \dfrac{4}{w-7} $ to get $ \dfrac{ \color{purple}{ 8w } }{ w^2-49 }$.

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}{ w-7 }$ and the second by $\color{blue}{ w+7 }$.

$$ \begin{aligned} \frac{4}{w+7} + \frac{4}{w-7} & = \frac{ 4 \cdot \color{blue}{ \left( w-7 \right) }}{ \left( w+7 \right) \cdot \color{blue}{ \left( w-7 \right) }} + \frac{ 4 \cdot \color{blue}{ \left( w+7 \right) }}{ \left( w-7 \right) \cdot \color{blue}{ \left( w+7 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 4w-28 } }{ w^2 -\cancel{7w}+ \cancel{7w}-49 } + \frac{ \color{purple}{ 4w+28 } }{ w^2 -\cancel{7w}+ \cancel{7w}-49 } = \\[1ex] &=\frac{ \color{purple}{ 8w } }{ w^2-49 } \end{aligned} $$

Divide $ \dfrac{w^2+3w}{w+7} $ by $ \dfrac{8w}{w^2-49} $ to get $ \dfrac{w^3-4w^2-21w}{8w} $.

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.

$$ \begin{aligned} \frac{ \frac{w^2+3w}{w+7} }{ \frac{\color{blue}{8w}}{\color{blue}{w^2-49}} } & \xlongequal{\text{Step 1}} \frac{w^2+3w}{w+7} \cdot \frac{\color{blue}{w^2-49}}{\color{blue}{8w}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ w^2+3w }{ 1 \cdot \color{red}{ \left( w+7 \right) } } \cdot \frac{ \left( w-7 \right) \cdot \color{red}{ \left( w+7 \right) } }{ 8w } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ w^2+3w }{ 1 } \cdot \frac{ w-7 }{ 8w } \xlongequal{\text{Step 4}} \frac{ \left( w^2+3w \right) \cdot \left( w-7 \right) }{ 1 \cdot 8w } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ w^3-7w^2+3w^2-21w }{ 8w } = \frac{w^3-4w^2-21w}{8w} \end{aligned} $$