Divide $ \, 2i \, $ by $ \, 2+i \, $ to get $\,\, \dfrac{2+4i}{5} $.
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Add $ \dfrac{2+4i}{5} $ and $ \dfrac{5}{2-i} $ to get $ \dfrac{ \color{purple}{ -4i^2+6i+29 } }{ -5i+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}{ -i+2 }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{2+4i}{5} + \frac{5}{2-i} & = \frac{ \left( 2+4i \right) \cdot \color{blue}{ \left( -i+2 \right) }}{ 5 \cdot \color{blue}{ \left( -i+2 \right) }} + \frac{ 5 \cdot \color{blue}{ 5 }}{ \left( 2-i \right) \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ -2i+4-4i^2+8i } }{ -5i+10 } + \frac{ \color{purple}{ 25 } }{ -5i+10 }=\frac{ \color{purple}{ -4i^2+6i+29 } }{ -5i+10 } \end{aligned} $$

$$ -4i^2 = -4 \cdot (-1) = 4 $$

Simplify numerator

$$ \color{blue}{4} +6i+ \color{blue}{29} = 6i+ \color{blue}{33} $$

Divide $ \, 33+6i \, $ by $ \, 10-5i \, $ to get $\,\, \dfrac{12+9i}{5} $.
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