Add $ \dfrac{2}{1+3i} $ and $ \dfrac{7}{-4i} $ to get $ \dfrac{ \color{purple}{ 13i+7 } }{ -12i^2-4i }$.

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}{ -4i }$ and the second by $\color{blue}{ 3i+1 }$.

$$ \begin{aligned} \frac{2}{1+3i} + \frac{7}{-4i} & = \frac{ 2 \cdot \color{blue}{ \left( -4i \right) }}{ \left( 1+3i \right) \cdot \color{blue}{ \left( -4i \right) }} + \frac{ 7 \cdot \color{blue}{ \left( 3i+1 \right) }}{ \left( -4i \right) \cdot \color{blue}{ \left( 3i+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ -8i } }{ -4i-12i^2 } + \frac{ \color{purple}{ 21i+7 } }{ -4i-12i^2 }=\frac{ \color{purple}{ 13i+7 } }{ -12i^2-4i } \end{aligned} $$

$$ -12i^2 = -12 \cdot (-1) = 12 $$

Divide $ \, 7+13i \, $ by $ \, 12-4i \, $ to get $\,\, \dfrac{4+23i}{20} $.
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