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

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.

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

Add $ \dfrac{x^2}{2} $ and $ 7x $ to get $ \dfrac{ \color{purple}{ x^2+14x } }{ 2 }$.

Step 1: Write $ 7x $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{x^2}{2} +7x & \xlongequal{\text{Step 1}} \frac{x^2}{2} + \frac{7x}{\color{red}{1}} = \frac{ x^2 }{ 2 } + \frac{ 7x \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2 } }{ 2 } + \frac{ \color{purple}{ 14x } }{ 2 }=\frac{ \color{purple}{ x^2+14x } }{ 2 } \end{aligned} $$

Subtract $3$ from $ \dfrac{x^2+14x}{2} $ to get $ \dfrac{ \color{purple}{ x^2+14x-6 } }{ 2 }$.

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

$$ \begin{aligned} \frac{x^2+14x}{2} -3 & \xlongequal{\text{Step 1}} \frac{x^2+14x}{2} - \frac{3}{\color{red}{1}} = \frac{ x^2+14x }{ 2 } - \frac{ 3 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2+14x } }{ 2 } - \frac{ \color{purple}{ 6 } }{ 2 }=\frac{ \color{purple}{ x^2+14x-6 } }{ 2 } \end{aligned} $$

Multiply $5$ by $ \dfrac{x^2+14x-6}{2} $ to get $ \dfrac{ 5x^2+70x-30 }{ 2 } $.

Step 1: Write $ 5 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 5 \cdot \frac{x^2+14x-6}{2} & \xlongequal{\text{Step 1}} \frac{5}{\color{red}{1}} \cdot \frac{x^2+14x-6}{2} \xlongequal{\text{Step 2}} \frac{ 5 \cdot \left( x^2+14x-6 \right) }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5x^2+70x-30 }{ 2 } \end{aligned} $$

Subtract $ \dfrac{5x^2+70x-30}{2} $ from $ -2x^2+8x+1 $ to get $ \dfrac{ \color{purple}{ -9x^2-54x+32 } }{ 2 }$.

Step 1: Write $ -2x^2+8x+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 first fraction by $\color{blue}{ 2 }$.

$$ \begin{aligned} -2x^2+8x+1- \frac{5x^2+70x-30}{2} & \xlongequal{\text{Step 1}} \frac{-2x^2+8x+1}{\color{red}{1}} - \frac{5x^2+70x-30}{2} = \frac{ \left( -2x^2+8x+1 \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} - \frac{ 5x^2+70x-30 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -4x^2+16x+2 } }{ 2 } - \frac{ \color{purple}{ 5x^2+70x-30 } }{ 2 }=\frac{ \color{purple}{ -4x^2+16x+2 - \left( 5x^2+70x-30 \right) } }{ 2 } = \\[1ex] &=\frac{ \color{purple}{ -9x^2-54x+32 } }{ 2 } \end{aligned} $$