Add $3i$ and $ \dfrac{1}{2-i} $ to get $ \dfrac{ \color{purple}{ -3i^2+6i+1 } }{ -i+2 }$.

Step 1: Write $ 3i $ 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 $\color{blue}{ -i+2 }$.

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

$$ -3i^2 = -3 \cdot (-1) = 3 $$

Simplify numerator

$$ \color{blue}{3} +6i+ \color{blue}{1} = 6i+ \color{blue}{4} $$

Divide $ \, 4+6i \, $ by $ \, 2-i \, $ to get $\,\, \dfrac{2+16i}{5} $.
( view steps )