Multiply $ \dfrac{v}{8} $ by $ v $ to get $ \dfrac{ v^2 }{ 8 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{v}{8} \cdot v & \xlongequal{\text{Step 1}} \frac{v}{8} \cdot \frac{v}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ v \cdot v }{ 8 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ v^2 }{ 8 } \end{aligned} $$

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

$$ \begin{aligned} 3 \cdot \frac{v}{8} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{v}{8} \xlongequal{\text{Step 2}} \frac{ 3 \cdot v }{ 1 \cdot 8 } \xlongequal{\text{Step 3}} \frac{ 3v }{ 8 } \end{aligned} $$

Subtract $ \dfrac{v^2}{8} $ from $ u $ to get $ \dfrac{ \color{purple}{ -v^2+8u } }{ 8 }$.

Step 1: Write $ u $ 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 $\color{blue}{ 8 }$.

$$ \begin{aligned} u- \frac{v^2}{8} & \xlongequal{\text{Step 1}} \frac{u}{\color{red}{1}} - \frac{v^2}{8} = \frac{ u \cdot \color{blue}{ 8 }}{ 1 \cdot \color{blue}{ 8 }} - \frac{ v^2 }{ 8 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 8u } }{ 8 } - \frac{ \color{purple}{ v^2 } }{ 8 }=\frac{ \color{purple}{ -v^2+8u } }{ 8 } \end{aligned} $$

Multiply $ \dfrac{3v}{8} $ by $ v $ to get $ \dfrac{ 3v^2 }{ 8 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3v}{8} \cdot v & \xlongequal{\text{Step 1}} \frac{3v}{8} \cdot \frac{v}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3v \cdot v }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3v^2 }{ 8 } \end{aligned} $$

Add $ \dfrac{-v^2+8u}{8} $ and $ 6u $ to get $ \dfrac{ \color{purple}{ -v^2+56u } }{ 8 }$.

Step 1: Write $ 6u $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} \frac{-v^2+8u}{8} +6u & \xlongequal{\text{Step 1}} \frac{-v^2+8u}{8} + \frac{6u}{\color{red}{1}} = \frac{ -v^2+8u }{ 8 } + \frac{ 6u \cdot \color{blue}{ 8 }}{ 1 \cdot \color{blue}{ 8 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -v^2+8u } }{ 8 } + \frac{ \color{purple}{ 48u } }{ 8 }=\frac{ \color{purple}{ -v^2+56u } }{ 8 } \end{aligned} $$

Multiply $ \dfrac{3v}{8} $ by $ v $ to get $ \dfrac{ 3v^2 }{ 8 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3v}{8} \cdot v & \xlongequal{\text{Step 1}} \frac{3v}{8} \cdot \frac{v}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3v \cdot v }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3v^2 }{ 8 } \end{aligned} $$

Subtract $ \dfrac{3v^2}{8} $ from $ \dfrac{-v^2+56u}{8} $ to get $ \dfrac{-4v^2+56u}{8} $.

To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{-v^2+56u}{8} - \frac{3v^2}{8} & = \frac{-v^2+56u}{\color{blue}{8}} - \frac{3v^2}{\color{blue}{8}} = \\[1ex] &=\frac{ -v^2+56u - 3v^2 }{ \color{blue}{ 8 }}= \frac{-4v^2+56u}{8} \end{aligned} $$