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

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

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

Subtract $ \dfrac{1}{2} $ from $ x $ to get $ \dfrac{ \color{purple}{ 2x-1 } }{ 2 }$.

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

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

Multiply $ \dfrac{2x-3}{2} $ by $ \dfrac{2x-1}{2} $ to get $ \dfrac{4x^2-8x+3}{4} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2x-3}{2} \cdot \frac{2x-1}{2} & \xlongequal{\text{Step 1}} \frac{ \left( 2x-3 \right) \cdot \left( 2x-1 \right) }{ 2 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 4x^2-2x-6x+3 }{ 4 } = \frac{4x^2-8x+3}{4} \end{aligned} $$

Multiply $ \dfrac{4x^2-8x+3}{4} $ by $ x-1 $ to get $ \dfrac{4x^3-12x^2+11x-3}{4} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4x^2-8x+3}{4} \cdot x-1 & \xlongequal{\text{Step 1}} \frac{4x^2-8x+3}{4} \cdot \frac{x-1}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 4x^2-8x+3 \right) \cdot \left( x-1 \right) }{ 4 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x^3-4x^2-8x^2+8x+3x-3 }{ 4 } = \frac{4x^3-12x^2+11x-3}{4} \end{aligned} $$