Multiply $ \dfrac{5-12i}{13} $ by $ 1+i $ to get $ \dfrac{-12i^2-7i+5}{13} $.

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.

$$ \begin{aligned} \frac{5-12i}{13} \cdot 1+i & \xlongequal{\text{Step 1}} \frac{5-12i}{13} \cdot \frac{1+i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 5-12i \right) \cdot \left( 1+i \right) }{ 13 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 5+5i-12i-12i^2 }{ 13 } = \frac{-12i^2-7i+5}{13} \end{aligned} $$

$$ -12i^2 = -12 \cdot (-1) = 12 $$

Combine like terms:

$$ \color{blue}{12} -7i+ \color{blue}{5} = -7i+ \color{blue}{17} $$

Subtract $ \dfrac{-7i+17}{13} $ from $ 4+i $ to get $ \dfrac{ \color{purple}{ 20i+35 } }{ 13 }$.

Step 1: Write $ 4+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 $\color{blue}{ 13 }$.

$$ \begin{aligned} 4+i- \frac{-7i+17}{13} & \xlongequal{\text{Step 1}} \frac{4+i}{\color{red}{1}} - \frac{-7i+17}{13} = \frac{ \left( 4+i \right) \cdot \color{blue}{ 13 }}{ 1 \cdot \color{blue}{ 13 }} - \frac{ -7i+17 }{ 13 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 52+13i } }{ 13 } - \frac{ \color{purple}{ -7i+17 } }{ 13 }=\frac{ \color{purple}{ 52+13i - \left( -7i+17 \right) } }{ 13 } = \\[1ex] &=\frac{ \color{purple}{ 20i+35 } }{ 13 } \end{aligned} $$