Multiply $i$ by $ \dfrac{2818+509i}{2665} $ to get $ \dfrac{509i^2+2818i}{2665} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} i \cdot \frac{2818+509i}{2665} & \xlongequal{\text{Step 1}} \frac{i}{\color{red}{1}} \cdot \frac{2818+509i}{2665} \xlongequal{\text{Step 2}} \frac{ i \cdot \left( 2818+509i \right) }{ 1 \cdot 2665 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2818i+509i^2 }{ 2665 } = \frac{509i^2+2818i}{2665} \end{aligned} $$

Multiply $2$ by $ \dfrac{254-113i}{205} $ to get $ \dfrac{-226i+508}{205} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{254-113i}{205} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{254-113i}{205} \xlongequal{\text{Step 2}} \frac{ 2 \cdot \left( 254-113i \right) }{ 1 \cdot 205 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 508-226i }{ 205 } = \frac{-226i+508}{205} \end{aligned} $$

$$ 509i^2 = 509 \cdot (-1) = -509 $$

Multiply $2$ by $ \dfrac{254-113i}{205} $ to get $ \dfrac{-226i+508}{205} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{254-113i}{205} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{254-113i}{205} \xlongequal{\text{Step 2}} \frac{ 2 \cdot \left( 254-113i \right) }{ 1 \cdot 205 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 508-226i }{ 205 } = \frac{-226i+508}{205} \end{aligned} $$

Add $1$ and $ \dfrac{-509+2818i}{2665} $ to get $ \dfrac{ \color{purple}{ 2818i+2156 } }{ 2665 }$.

Step 1: Write $ 1 $ 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 $\color{blue}{ 2665 }$.

$$ \begin{aligned} 1+ \frac{-509+2818i}{2665} & \xlongequal{\text{Step 1}} \frac{1}{\color{red}{1}} + \frac{-509+2818i}{2665} = \frac{ 1 \cdot \color{blue}{ 2665 }}{ 1 \cdot \color{blue}{ 2665 }} + \frac{ -509+2818i }{ 2665 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2665 } }{ 2665 } + \frac{ \color{purple}{ -509+2818i } }{ 2665 }=\frac{ \color{purple}{ 2818i+2156 } }{ 2665 } \end{aligned} $$

Multiply $2$ by $ \dfrac{254-113i}{205} $ to get $ \dfrac{-226i+508}{205} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{254-113i}{205} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{254-113i}{205} \xlongequal{\text{Step 2}} \frac{ 2 \cdot \left( 254-113i \right) }{ 1 \cdot 205 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 508-226i }{ 205 } = \frac{-226i+508}{205} \end{aligned} $$

Subtract $ \dfrac{-226i+508}{205} $ from $ \dfrac{2818i+2156}{2665} $ to get $ \dfrac{ \color{purple}{ 5756i-4448 } }{ 2665 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{13}$.

$$ \begin{aligned} \frac{2818i+2156}{2665} - \frac{-226i+508}{205} & = \frac{ 2818i+2156 }{ 2665 } - \frac{ \left( -226i+508 \right) \cdot \color{blue}{ 13 }}{ 205 \cdot \color{blue}{ 13 }} = \\[1ex] &=\frac{ \color{purple}{ 2818i+2156 } }{ 2665 } - \frac{ \color{purple}{ -2938i+6604 } }{ 2665 }=\frac{ \color{purple}{ 2818i+2156 - \left( -2938i+6604 \right) } }{ 2665 } = \\[1ex] &=\frac{ \color{purple}{ 5756i-4448 } }{ 2665 } \end{aligned} $$