Divide $ \, 1-2i \, $ by $ \, 3+4i \, $ to get $\,\, \dfrac{-1-2i}{5} $.
( view steps )

Add $3i$ and $ \dfrac{-1-2i}{5} $ to get $ \dfrac{ \color{purple}{ 13i-1 } }{ 5 }$.

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 $\color{blue}{ 5 }$.

$$ \begin{aligned} 3i+ \frac{-1-2i}{5} & \xlongequal{\text{Step 1}} \frac{3i}{\color{red}{1}} + \frac{-1-2i}{5} = \frac{ 3i \cdot \color{blue}{ 5 }}{ 1 \cdot \color{blue}{ 5 }} + \frac{ -1-2i }{ 5 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 15i } }{ 5 } + \frac{ \color{purple}{ -1-2i } }{ 5 }=\frac{ \color{purple}{ 13i-1 } }{ 5 } \end{aligned} $$