Multiply $ \dfrac{5-12i}{13} $ by $ -1-2i $ to get $ \dfrac{24i^2+2i-5}{13} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{5-12i}{13} \cdot -1-2i & \xlongequal{\text{Step 1}} \frac{5-12i}{13} \cdot \frac{-1-2i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 5-12i \right) \cdot \left( -1-2i \right) }{ 13 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -5-10i+12i+24i^2 }{ 13 } = \frac{24i^2+2i-5}{13} \end{aligned} $$

$$ 24i^2 = 24 \cdot (-1) = -24 $$

Combine like terms:

$$ \color{blue}{-24} +2i \color{blue}{-5} = 2i \color{blue}{-29} $$

Subtract $ \dfrac{2i-29}{13} $ from $ 1+i $ to get $ \dfrac{ \color{purple}{ 11i+42 } }{ 13 }$.

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

$$ \begin{aligned} 1+i- \frac{2i-29}{13} & \xlongequal{\text{Step 1}} \frac{1+i}{\color{red}{1}} - \frac{2i-29}{13} = \frac{ \left( 1+i \right) \cdot \color{blue}{ 13 }}{ 1 \cdot \color{blue}{ 13 }} - \frac{ 2i-29 }{ 13 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 13+13i } }{ 13 } - \frac{ \color{purple}{ 2i-29 } }{ 13 }=\frac{ \color{purple}{ 13+13i - \left( 2i-29 \right) } }{ 13 } = \\[1ex] &=\frac{ \color{purple}{ 11i+42 } }{ 13 } \end{aligned} $$