Multiply $14$ by $ \dfrac{x}{2} $ to get $ \dfrac{ 14x }{ 2 } $.

Step 1: Write $ 14 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $21$ by $ \dfrac{y}{2} $ to get $ \dfrac{ 21y }{ 2 } $.

Step 1: Write $ 21 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 21 \cdot \frac{y}{2} & \xlongequal{\text{Step 1}} \frac{21}{\color{red}{1}} \cdot \frac{y}{2} \xlongequal{\text{Step 2}} \frac{ 21 \cdot y }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 21y }{ 2 } \end{aligned} $$

Multiply $ \dfrac{14x}{2} $ by $ x $ to get $ \dfrac{ 14x^2 }{ 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.

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

Multiply $ \dfrac{21y}{2} $ by $ x $ to get $ \dfrac{ 21xy }{ 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.

$$ \begin{aligned} \frac{21y}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{21y}{2} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 21y \cdot x }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 21xy }{ 2 } \end{aligned} $$

Add $ \dfrac{14x^2}{2} $ and $ 3y $ to get $ \dfrac{ \color{purple}{ 14x^2+6y } }{ 2 }$.

Step 1: Write $ 3y $ 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 $\color{blue}{2}$.

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

Multiply $ \dfrac{21y}{2} $ by $ x $ to get $ \dfrac{ 21xy }{ 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.

$$ \begin{aligned} \frac{21y}{2} \cdot x & \xlongequal{\text{Step 1}} \frac{21y}{2} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 21y \cdot x }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 21xy }{ 2 } \end{aligned} $$

Add $ \dfrac{14x^2+6y}{2} $ and $ \dfrac{21xy}{2} $ to get $ \dfrac{14x^2+21xy+6y}{2} $.

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

$$ \begin{aligned} \frac{14x^2+6y}{2} + \frac{21xy}{2} & = \frac{14x^2+6y}{\color{blue}{2}} + \frac{21xy}{\color{blue}{2}} = \\[1ex] &=\frac{ 14x^2+6y + 21xy }{ \color{blue}{ 2 }}= \frac{14x^2+21xy+6y}{2} \end{aligned} $$

Add $ \dfrac{14x^2+21xy+6y}{2} $ and $ 3y $ to get $ \dfrac{ \color{purple}{ 14x^2+21xy+12y } }{ 2 }$.

Step 1: Write $ 3y $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{14x^2+21xy+6y}{2} +3y & \xlongequal{\text{Step 1}} \frac{14x^2+21xy+6y}{2} + \frac{3y}{\color{red}{1}} = \frac{ 14x^2+21xy+6y }{ 2 } + \frac{ 3y \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 14x^2+21xy+6y } }{ 2 } + \frac{ \color{purple}{ 6y } }{ 2 }=\frac{ \color{purple}{ 14x^2+21xy+12y } }{ 2 } \end{aligned} $$