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.

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

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

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.

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

Multiply $ \dfrac{3}{8} $ by $ x^2 $ to get $ \dfrac{ 3x^2 }{ 8 } $.

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.

$$ \begin{aligned} \frac{3}{8} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{3}{8} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x^2 }{ 8 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3x^2 }{ 8 } \end{aligned} $$

Multiply $ \dfrac{3}{4} $ by $ x $ to get $ \dfrac{ 3x }{ 4 } $.

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.

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

Add $ \dfrac{x^2}{2} $ and $ \dfrac{2x}{3} $ to get $ \dfrac{ \color{purple}{ 3x^2+4x } }{ 6 }$.

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}{ 3 }$ and the second by $\color{blue}{ 2 }$.

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

Subtract $ \dfrac{3x}{4} $ from $ \dfrac{3x^2}{8} $ to get $ \dfrac{ \color{purple}{ 3x^2-6x } }{ 8 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{3x^2}{8} - \frac{3x}{4} & = \frac{ 3x^2 }{ 8 } - \frac{ 3x \cdot \color{blue}{ 2 }}{ 4 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3x^2 } }{ 8 } - \frac{ \color{purple}{ 6x } }{ 8 }=\frac{ \color{purple}{ 3x^2-6x } }{ 8 } \end{aligned} $$

Subtract $ \dfrac{5}{3} $ from $ \dfrac{3x^2+4x}{6} $ to get $ \dfrac{ \color{purple}{ 3x^2+4x-10 } }{ 6 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{3x^2+4x}{6} - \frac{5}{3} & = \frac{ 3x^2+4x }{ 6 } - \frac{ 5 \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 3x^2+4x } }{ 6 } - \frac{ \color{purple}{ 10 } }{ 6 }=\frac{ \color{purple}{ 3x^2+4x-10 } }{ 6 } \end{aligned} $$

Add $ \dfrac{3x^2-6x}{8} $ and $ \dfrac{7}{3} $ to get $ \dfrac{ \color{purple}{ 9x^2-18x+56 } }{ 24 }$.

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}{ 3 }$ and the second by $\color{blue}{ 8 }$.

$$ \begin{aligned} \frac{3x^2-6x}{8} + \frac{7}{3} & = \frac{ \left( 3x^2-6x \right) \cdot \color{blue}{ 3 }}{ 8 \cdot \color{blue}{ 3 }} + \frac{ 7 \cdot \color{blue}{ 8 }}{ 3 \cdot \color{blue}{ 8 }} = \\[1ex] &=\frac{ \color{purple}{ 9x^2-18x } }{ 24 } + \frac{ \color{purple}{ 56 } }{ 24 }=\frac{ \color{purple}{ 9x^2-18x+56 } }{ 24 } \end{aligned} $$

Subtract $ \dfrac{9x^2-18x+56}{24} $ from $ \dfrac{3x^2+4x-10}{6} $ to get $ \dfrac{ \color{purple}{ 3x^2+34x-96 } }{ 24 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 4 }$.

$$ \begin{aligned} \frac{3x^2+4x-10}{6} - \frac{9x^2-18x+56}{24} & = \frac{ \left( 3x^2+4x-10 \right) \cdot \color{blue}{ 4 }}{ 6 \cdot \color{blue}{ 4 }} - \frac{ 9x^2-18x+56 }{ 24 } = \\[1ex] &=\frac{ \color{purple}{ 12x^2+16x-40 } }{ 24 } - \frac{ \color{purple}{ 9x^2-18x+56 } }{ 24 }=\frac{ \color{purple}{ 12x^2+16x-40 - \left( 9x^2-18x+56 \right) } }{ 24 } = \\[1ex] &=\frac{ \color{purple}{ 3x^2+34x-96 } }{ 24 } \end{aligned} $$