Divide $ \, 3+4i \, $ by $ \, 2+5i \, $ to get $\,\, \dfrac{26-7i}{29} $.
( view steps )

Subtract $2i$ from $ \dfrac{26-7i}{29} $ to get $ \dfrac{ \color{purple}{ -65i+26 } }{ 29 }$.

Step 1: Write $ 2i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{29}$.

$$ \begin{aligned} \frac{26-7i}{29} -2i & \xlongequal{\text{Step 1}} \frac{26-7i}{29} - \frac{2i}{\color{red}{1}} = \frac{ 26-7i }{ 29 } - \frac{ 2i \cdot \color{blue}{ 29 }}{ 1 \cdot \color{blue}{ 29 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 26-7i } }{ 29 } - \frac{ \color{purple}{ 58i } }{ 29 }=\frac{ \color{purple}{ -65i+26 } }{ 29 } \end{aligned} $$

Add $ \dfrac{-65i+26}{29} $ and $ 3 $ to get $ \dfrac{ \color{purple}{ -65i+113 } }{ 29 }$.

Step 1: Write $ 3 $ 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 second fraction by $\color{blue}{29}$.

$$ \begin{aligned} \frac{-65i+26}{29} +3 & \xlongequal{\text{Step 1}} \frac{-65i+26}{29} + \frac{3}{\color{red}{1}} = \frac{ -65i+26 }{ 29 } + \frac{ 3 \cdot \color{blue}{ 29 }}{ 1 \cdot \color{blue}{ 29 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -65i+26 } }{ 29 } + \frac{ \color{purple}{ 87 } }{ 29 }=\frac{ \color{purple}{ -65i+113 } }{ 29 } \end{aligned} $$