Multiply $42$ by $ \dfrac{v^2}{2} $ to get $ \dfrac{ 42v^2 }{ 2 } $.

Step 1: Write $ 42 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 42 \cdot \frac{v^2}{2} & \xlongequal{\text{Step 1}} \frac{42}{\color{red}{1}} \cdot \frac{v^2}{2} \xlongequal{\text{Step 2}} \frac{ 42 \cdot v^2 }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 42v^2 }{ 2 } \end{aligned} $$

Multiply $ \dfrac{42v^2}{2} $ by $ v^2 $ to get $ \dfrac{ 42v^4 }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{42v^2}{2} \cdot v^2 & \xlongequal{\text{Step 1}} \frac{42v^2}{2} \cdot \frac{v^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 42v^2 \cdot v^2 }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 42v^4 }{ 2 } \end{aligned} $$

Add $6v^3$ and $ \dfrac{42v^4}{2} $ to get $ \dfrac{ \color{purple}{ 42v^4+12v^3 } }{ 2 }$.

Step 1: Write $ 6v^3 $ 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 first fraction by $\color{blue}{ 2 }$.

$$ \begin{aligned} 6v^3+ \frac{42v^4}{2} & \xlongequal{\text{Step 1}} \frac{6v^3}{\color{red}{1}} + \frac{42v^4}{2} = \frac{ 6v^3 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 42v^4 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 12v^3 } }{ 2 } + \frac{ \color{purple}{ 42v^4 } }{ 2 }=\frac{ \color{purple}{ 42v^4+12v^3 } }{ 2 } \end{aligned} $$

Add $ \dfrac{42v^4+12v^3}{2} $ and $ 26v $ to get $ \dfrac{ \color{purple}{ 42v^4+12v^3+52v } }{ 2 }$.

Step 1: Write $ 26v $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{42v^4+12v^3}{2} +26v & \xlongequal{\text{Step 1}} \frac{42v^4+12v^3}{2} + \frac{26v}{\color{red}{1}} = \frac{ 42v^4+12v^3 }{ 2 } + \frac{ 26v \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 42v^4+12v^3 } }{ 2 } + \frac{ \color{purple}{ 52v } }{ 2 }=\frac{ \color{purple}{ 42v^4+12v^3+52v } }{ 2 } \end{aligned} $$

Add $ \dfrac{42v^4+12v^3+52v}{2} $ and $ 84 $ to get $ \dfrac{ \color{purple}{ 42v^4+12v^3+52v+168 } }{ 2 }$.

Step 1: Write $ 84 $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{42v^4+12v^3+52v}{2} +84 & \xlongequal{\text{Step 1}} \frac{42v^4+12v^3+52v}{2} + \frac{84}{\color{red}{1}} = \frac{ 42v^4+12v^3+52v }{ 2 } + \frac{ 84 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 42v^4+12v^3+52v } }{ 2 } + \frac{ \color{purple}{ 168 } }{ 2 }=\frac{ \color{purple}{ 42v^4+12v^3+52v+168 } }{ 2 } \end{aligned} $$