Divide both the top and bottom numbers by $ \color{orangered}{ 2 } $.

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

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

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

Subtract $1$ from $ \dfrac{-4x^2+x}{2} $ to get $ \dfrac{ \color{purple}{ -4x^2+x-2 } }{ 2 }$.

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

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