Multiply $2$ by $ \dfrac{2818+509i}{2665} $ to get $ \dfrac{1018i+5636}{2665} $.

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.

$$ \begin{aligned} 2 \cdot \frac{2818+509i}{2665} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{2818+509i}{2665} \xlongequal{\text{Step 2}} \frac{ 2 \cdot \left( 2818+509i \right) }{ 1 \cdot 2665 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5636+1018i }{ 2665 } = \frac{1018i+5636}{2665} \end{aligned} $$

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

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.

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

Subtract $ \dfrac{1018i+5636}{2665} $ from $ 1 $ to get $ \dfrac{ \color{purple}{ -1018i-2971 } }{ 2665 }$.

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

$$ \begin{aligned} 1- \frac{1018i+5636}{2665} & \xlongequal{\text{Step 1}} \frac{1}{\color{red}{1}} - \frac{1018i+5636}{2665} = \frac{ 1 \cdot \color{blue}{ 2665 }}{ 1 \cdot \color{blue}{ 2665 }} - \frac{ 1018i+5636 }{ 2665 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2665 } }{ 2665 } - \frac{ \color{purple}{ 1018i+5636 } }{ 2665 }=\frac{ \color{purple}{ 2665 - \left( 1018i+5636 \right) } }{ 2665 } = \\[1ex] &=\frac{ \color{purple}{ -1018i-2971 } }{ 2665 } \end{aligned} $$

$$ -113i^2 = -113 \cdot (-1) = 113 $$

Add $ \dfrac{-1018i-2971}{2665} $ and $ \dfrac{113+254i}{205} $ to get $ \dfrac{ \color{purple}{ 2284i-1502 } }{ 2665 }$.

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

$$ \begin{aligned} \frac{-1018i-2971}{2665} + \frac{113+254i}{205} & = \frac{ -1018i-2971 }{ 2665 } + \frac{ \left( 113+254i \right) \cdot \color{blue}{ 13 }}{ 205 \cdot \color{blue}{ 13 }} = \\[1ex] &=\frac{ \color{purple}{ -1018i-2971 } }{ 2665 } + \frac{ \color{purple}{ 1469+3302i } }{ 2665 }=\frac{ \color{purple}{ 2284i-1502 } }{ 2665 } \end{aligned} $$