Multiply $3$ by $ \dfrac{x-1}{2} $ to get $ \dfrac{ 3x-3 }{ 2 } $.

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.

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

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

Step 1: Write $ 2x^3 $ 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 second fraction by $\color{blue}{2}$.

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

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

Step 1: Write $ 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 second fraction by $\color{blue}{2}$.

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

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

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

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

Add $ \dfrac{-4x^3+2x^2+3x+3}{2} $ and $ 2x $ to get $ \dfrac{ \color{purple}{ -4x^3+2x^2+7x+3 } }{ 2 }$.

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

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