Divide $ \, i \, $ by $ \, 1-3i \, $ to get $\,\, \dfrac{-3+i}{10} $.
( view steps )

Multiply $4$ by $ \dfrac{-3+i}{10} $ to get $ \dfrac{4i-12}{10} $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{-3+i}{10} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{-3+i}{10} \xlongequal{\text{Step 2}} \frac{ 4 \cdot \left( -3+i \right) }{ 1 \cdot 10 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -12+4i }{ 10 } = \frac{4i-12}{10} \end{aligned} $$

Add $ \dfrac{3}{1+2i} $ and $ \dfrac{4i-12}{10} $ to get $ \dfrac{ \color{purple}{ 8i^2-20i+18 } }{ 20i+10 }$.

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}{ 10 }$ and the second by $\color{blue}{ 2i+1 }$.

$$ \begin{aligned} \frac{3}{1+2i} + \frac{4i-12}{10} & = \frac{ 3 \cdot \color{blue}{ 10 }}{ \left( 1+2i \right) \cdot \color{blue}{ 10 }} + \frac{ \left( 4i-12 \right) \cdot \color{blue}{ \left( 2i+1 \right) }}{ 10 \cdot \color{blue}{ \left( 2i+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 30 } }{ 10+20i } + \frac{ \color{purple}{ 8i^2+4i-24i-12 } }{ 10+20i }=\frac{ \color{purple}{ 8i^2-20i+18 } }{ 20i+10 } \end{aligned} $$

$$ 8i^2 = 8 \cdot (-1) = -8 $$

Simplify numerator

$$ \color{blue}{-8} -20i+ \color{blue}{18} = -20i+ \color{blue}{10} $$

Divide $ \, 10-20i \, $ by $ \, 10+20i \, $ to get $\,\, \dfrac{-3-4i}{5} $.
( view steps )