Multiply $4$ by $ \dfrac{x^3}{6} $ to get $ \dfrac{ 4x^3 }{ 6 } $.

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.

$$ \begin{aligned} 4 \cdot \frac{x^3}{6} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{x^3}{6} \xlongequal{\text{Step 2}} \frac{ 4 \cdot x^3 }{ 1 \cdot 6 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x^3 }{ 6 } \end{aligned} $$

Multiply $10x-2x^2$ by $ \dfrac{x}{2} $ to get $ \dfrac{-2x^3+10x^2}{2} $.

Step 1: Write $ 10x-2x^2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 10x-2x^2 \cdot \frac{x}{2} & \xlongequal{\text{Step 1}} \frac{10x-2x^2}{\color{red}{1}} \cdot \frac{x}{2} \xlongequal{\text{Step 2}} \frac{ \left( 10x-2x^2 \right) \cdot x }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 10x^2-2x^3 }{ 2 } = \frac{-2x^3+10x^2}{2} \end{aligned} $$

Add $ \dfrac{4x^3}{6} $ and $ \dfrac{-2x^3+10x^2}{2} $ to get $ \dfrac{ \color{purple}{ -2x^3+30x^2 } }{ 6 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{4x^3}{6} + \frac{-2x^3+10x^2}{2} & = \frac{ 4x^3 }{ 6 } + \frac{ \left( -2x^3+10x^2 \right) \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 4x^3 } }{ 6 } + \frac{ \color{purple}{ -6x^3+30x^2 } }{ 6 }=\frac{ \color{purple}{ -2x^3+30x^2 } }{ 6 } \end{aligned} $$

Add $ \dfrac{-2x^3+30x^2}{6} $ and $ 37x $ to get $ \dfrac{ \color{purple}{ -2x^3+30x^2+222x } }{ 6 }$.

Step 1: Write $ 37x $ 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 $\color{blue}{6}$.

$$ \begin{aligned} \frac{-2x^3+30x^2}{6} +37x & \xlongequal{\text{Step 1}} \frac{-2x^3+30x^2}{6} + \frac{37x}{\color{red}{1}} = \frac{ -2x^3+30x^2 }{ 6 } + \frac{ 37x \cdot \color{blue}{ 6 }}{ 1 \cdot \color{blue}{ 6 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -2x^3+30x^2 } }{ 6 } + \frac{ \color{purple}{ 222x } }{ 6 }=\frac{ \color{purple}{ -2x^3+30x^2+222x } }{ 6 } \end{aligned} $$

Subtract $ \dfrac{250}{3} $ from $ \dfrac{-2x^3+30x^2+222x}{6} $ to get $ \dfrac{ \color{purple}{ -2x^3+30x^2+222x-500 } }{ 6 }$.

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

$$ \begin{aligned} \frac{-2x^3+30x^2+222x}{6} - \frac{250}{3} & = \frac{ -2x^3+30x^2+222x }{ 6 } - \frac{ 250 \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ -2x^3+30x^2+222x } }{ 6 } - \frac{ \color{purple}{ 500 } }{ 6 }=\frac{ \color{purple}{ -2x^3+30x^2+222x-500 } }{ 6 } \end{aligned} $$