Divide $ \, 1+3i \, $ by $ \, 2+3i \, $ to get $\,\, \dfrac{11+3i}{13} $.
( view steps )

Subtract $2i+2$ from $ \dfrac{11+3i}{13} $ to get $ \dfrac{ \color{purple}{ -23i-15 } }{ 13 }$.

Step 1: Write $ 2i+2 $ 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 $\color{blue}{13}$.

$$ \begin{aligned} \frac{11+3i}{13} -2i+2 & \xlongequal{\text{Step 1}} \frac{11+3i}{13} - \frac{2i+2}{\color{red}{1}} = \frac{ 11+3i }{ 13 } - \frac{ \left( 2i+2 \right) \cdot \color{blue}{ 13 }}{ 1 \cdot \color{blue}{ 13 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 11+3i } }{ 13 } - \frac{ \color{purple}{ 26i+26 } }{ 13 }=\frac{ \color{purple}{ 11+3i - \left( 26i+26 \right) } }{ 13 } = \\[1ex] &=\frac{ \color{purple}{ -23i-15 } }{ 13 } \end{aligned} $$