Multiply $ \dfrac{1}{4} $ by $ x^2 $ to get $ \dfrac{ x^2 }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{1}{4} $ from $ 0x^2+2x $ to get $ \dfrac{ \color{purple}{ 8x-1 } }{ 4 }$.

Step 1: Write $ 0x^2+2x $ 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 $\color{blue}{ 4 }$.

$$ \begin{aligned} 0x^2+2x- \frac{1}{4} & \xlongequal{\text{Step 1}} \frac{0x^2+2x}{\color{red}{1}} - \frac{1}{4} = \frac{ \left( 0x^2+2x \right) \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} - \frac{ 1 }{ 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 0x^2+8x } }{ 4 } - \frac{ \color{purple}{ 1 } }{ 4 }=\frac{ \color{purple}{ 8x-1 } }{ 4 } \end{aligned} $$

Subtract $3x$ from $ \dfrac{x^2}{4} $ to get $ \dfrac{ \color{purple}{ x^2-12x } }{ 4 }$.

Step 1: Write $ 3x $ 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 $\color{blue}{4}$.

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

Subtract $ \dfrac{1}{4} $ from $ 0x^2+2x $ to get $ \dfrac{ \color{purple}{ 8x-1 } }{ 4 }$.

Step 1: Write $ 0x^2+2x $ 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 $\color{blue}{ 4 }$.

$$ \begin{aligned} 0x^2+2x- \frac{1}{4} & \xlongequal{\text{Step 1}} \frac{0x^2+2x}{\color{red}{1}} - \frac{1}{4} = \frac{ \left( 0x^2+2x \right) \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} - \frac{ 1 }{ 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 0x^2+8x } }{ 4 } - \frac{ \color{purple}{ 1 } }{ 4 }=\frac{ \color{purple}{ 8x-1 } }{ 4 } \end{aligned} $$

Add $ \dfrac{x^2-12x}{4} $ and $ 2 $ to get $ \dfrac{ \color{purple}{ x^2-12x+8 } }{ 4 }$.

Step 1: Write $ 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 $\color{blue}{4}$.

$$ \begin{aligned} \frac{x^2-12x}{4} +2 & \xlongequal{\text{Step 1}} \frac{x^2-12x}{4} + \frac{2}{\color{red}{1}} = \frac{ x^2-12x }{ 4 } + \frac{ 2 \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2-12x } }{ 4 } + \frac{ \color{purple}{ 8 } }{ 4 }=\frac{ \color{purple}{ x^2-12x+8 } }{ 4 } \end{aligned} $$

Subtract $ \dfrac{1}{4} $ from $ 0x^2+2x $ to get $ \dfrac{ \color{purple}{ 8x-1 } }{ 4 }$.

Step 1: Write $ 0x^2+2x $ 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 $\color{blue}{ 4 }$.

$$ \begin{aligned} 0x^2+2x- \frac{1}{4} & \xlongequal{\text{Step 1}} \frac{0x^2+2x}{\color{red}{1}} - \frac{1}{4} = \frac{ \left( 0x^2+2x \right) \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} - \frac{ 1 }{ 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 0x^2+8x } }{ 4 } - \frac{ \color{purple}{ 1 } }{ 4 }=\frac{ \color{purple}{ 8x-1 } }{ 4 } \end{aligned} $$

Add $ \dfrac{x^2-12x+8}{4} $ and $ \dfrac{8x-1}{4} $ to get $ \dfrac{x^2-4x+7}{4} $.

To add expressions with the same denominators, we add the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{x^2-12x+8}{4} + \frac{8x-1}{4} & = \frac{x^2-12x+8}{\color{blue}{4}} + \frac{8x-1}{\color{blue}{4}} = \\[1ex] &=\frac{ x^2-12x+8 + \left( 8x-1 \right) }{ \color{blue}{ 4 }}= \frac{x^2-4x+7}{4} \end{aligned} $$