Add $ \dfrac{x}{1+i} $ and $ \dfrac{y}{1-2i} $ to get $ \dfrac{ \color{purple}{ -2ix+iy+x+y } }{ -2i^2-i+1 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ -2i+1 }$ and the second by $\color{blue}{ i+1 }$.

$$ \begin{aligned} \frac{x}{1+i} + \frac{y}{1-2i} & = \frac{ x \cdot \color{blue}{ \left( -2i+1 \right) }}{ \left( 1+i \right) \cdot \color{blue}{ \left( -2i+1 \right) }} + \frac{ y \cdot \color{blue}{ \left( i+1 \right) }}{ \left( 1-2i \right) \cdot \color{blue}{ \left( i+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ -2ix+x } }{ -2i+1-2i^2+i } + \frac{ \color{purple}{ iy+y } }{ -2i+1-2i^2+i }=\frac{ \color{purple}{ -2ix+iy+x+y } }{ -2i^2-i+1 } \end{aligned} $$

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

Combine like terms:

$$ \color{blue}{2} -i+ \color{blue}{1} = -i+ \color{blue}{3} $$

Subtract $1$ from $ \dfrac{-2ix+iy+x+y}{-i+3} $ to get $ \dfrac{ \color{purple}{ -2ix+iy+i+x+y-3 } }{ -i+3 }$.

Step 1: Write $ 1 $ 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 $\color{blue}{-i+3}$.

$$ \begin{aligned} \frac{-2ix+iy+x+y}{-i+3} -1 & \xlongequal{\text{Step 1}} \frac{-2ix+iy+x+y}{-i+3} - \frac{1}{\color{red}{1}} = \frac{ -2ix+iy+x+y }{ -i+3 } - \frac{ 1 \cdot \color{blue}{ \left( -i+3 \right) }}{ 1 \cdot \color{blue}{ \left( -i+3 \right) }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -2ix+iy+x+y } }{ -i+3 } - \frac{ \color{purple}{ -i+3 } }{ -i+3 }=\frac{ \color{purple}{ -2ix+iy+x+y - \left( -i+3 \right) } }{ -i+3 } = \\[1ex] &=\frac{ \color{purple}{ -2ix+iy+i+x+y-3 } }{ -i+3 } \end{aligned} $$