Divide $ \dfrac{74-27i}{51} $ by $ -3-2i $ to get $ \dfrac{-27i+74}{-102i-153} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{74-27i}{51} }{-3-2i} & \xlongequal{\text{Step 1}} \frac{74-27i}{51} \cdot \frac{\color{blue}{1}}{\color{blue}{-3-2i}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 74-27i \right) \cdot 1 }{ 51 \cdot \left( -3-2i \right) } \xlongequal{\text{Step 3}} \frac{ 74-27i }{ -153-102i } = \\[1ex] &= \frac{-27i+74}{-102i-153} \end{aligned} $$

Divide $ \, 74-27i \, $ by $ \, -153-102i \, $ to get $\,\, \dfrac{-168+229i}{663} $.
( view steps )

Multiply $1+i$ by $ \dfrac{-168+229i}{663} $ to get $ \dfrac{229i^2+61i-168}{663} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 1+i \cdot \frac{-168+229i}{663} & \xlongequal{\text{Step 1}} \frac{1+i}{\color{red}{1}} \cdot \frac{-168+229i}{663} \xlongequal{\text{Step 2}} \frac{ \left( 1+i \right) \cdot \left( -168+229i \right) }{ 1 \cdot 663 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -168+229i-168i+229i^2 }{ 663 } = \frac{229i^2+61i-168}{663} \end{aligned} $$

$$ 229i^2 = 229 \cdot (-1) = -229 $$

Combine like terms:

$$ \color{blue}{-229} +61i \color{blue}{-168} = 61i \color{blue}{-397} $$

Subtract $ \dfrac{61i-397}{663} $ from $ -1-2i $ to get $ \dfrac{ \color{purple}{ -1387i-266 } }{ 663 }$.

Step 1: Write $ -1-2i $ 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 first fraction by $\color{blue}{ 663 }$.

$$ \begin{aligned} -1-2i- \frac{61i-397}{663} & \xlongequal{\text{Step 1}} \frac{-1-2i}{\color{red}{1}} - \frac{61i-397}{663} = \frac{ \left( -1-2i \right) \cdot \color{blue}{ 663 }}{ 1 \cdot \color{blue}{ 663 }} - \frac{ 61i-397 }{ 663 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -663-1326i } }{ 663 } - \frac{ \color{purple}{ 61i-397 } }{ 663 }=\frac{ \color{purple}{ -663-1326i - \left( 61i-397 \right) } }{ 663 } = \\[1ex] &=\frac{ \color{purple}{ -1387i-266 } }{ 663 } \end{aligned} $$