Multiply each term of $ \left( \color{blue}{2-i}\right) $ by each term in $ \left( 2+3i\right) $.

$$ \left( \color{blue}{2-i}\right) \cdot \left( 2+3i\right) = 4+6i-2i-3i^2 $$

Combine like terms:

$$ 4+ \color{blue}{6i} \color{blue}{-2i} -3i^2 = -3i^2+ \color{blue}{4i} +4 $$

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

Combine like terms:

$$ \color{blue}{3} +4i+ \color{blue}{4} = 4i+ \color{blue}{7} $$

Divide $ \, 1+2i \, $ by $ \, 7+4i \, $ to get $\,\, \dfrac{3+2i}{13} $.
( view steps )

Multiply $2i$ by $ \dfrac{3+2i}{13} $ to get $ \dfrac{4i^2+6i}{13} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2i \cdot \frac{3+2i}{13} & \xlongequal{\text{Step 1}} \frac{2i}{\color{red}{1}} \cdot \frac{3+2i}{13} \xlongequal{\text{Step 2}} \frac{ 2i \cdot \left( 3+2i \right) }{ 1 \cdot 13 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6i+4i^2 }{ 13 } = \frac{4i^2+6i}{13} \end{aligned} $$

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

Subtract $ \dfrac{-4+6i}{13} $ from $ 3 $ to get $ \dfrac{ \color{purple}{ -6i+43 } }{ 13 }$.

Step 1: Write $ 3 $ 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 first fraction by $\color{blue}{ 13 }$.

$$ \begin{aligned} 3- \frac{-4+6i}{13} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} - \frac{-4+6i}{13} = \frac{ 3 \cdot \color{blue}{ 13 }}{ 1 \cdot \color{blue}{ 13 }} - \frac{ -4+6i }{ 13 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 39 } }{ 13 } - \frac{ \color{purple}{ -4+6i } }{ 13 }=\frac{ \color{purple}{ 39 - \left( -4+6i \right) } }{ 13 } = \\[1ex] &=\frac{ \color{purple}{ -6i+43 } }{ 13 } \end{aligned} $$