Add $ \dfrac{4}{2-3i} $ and $ \dfrac{2}{1+i} $ to get $ \dfrac{ \color{purple}{ -2i+8 } }{ -3i^2-i+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+1 }$ and the second by $\color{blue}{ -3i+2 }$.

$$ \begin{aligned} \frac{4}{2-3i} + \frac{2}{1+i} & = \frac{ 4 \cdot \color{blue}{ \left( i+1 \right) }}{ \left( 2-3i \right) \cdot \color{blue}{ \left( i+1 \right) }} + \frac{ 2 \cdot \color{blue}{ \left( -3i+2 \right) }}{ \left( 1+i \right) \cdot \color{blue}{ \left( -3i+2 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 4i+4 } }{ 2i+2-3i^2-3i } + \frac{ \color{purple}{ -6i+4 } }{ 2i+2-3i^2-3i }=\frac{ \color{purple}{ -2i+8 } }{ -3i^2-i+2 } \end{aligned} $$

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

Simplify denominator

$$ \color{blue}{3} -i+ \color{blue}{2} = -i+ \color{blue}{5} $$

Divide $ \, 8-2i \, $ by $ \, 5-i \, $ to get $\,\, \dfrac{21-i}{13} $.
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