Multiply $2$ by $ \dfrac{i}{z^2} $ to get $ \dfrac{ 2i }{ z^2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{i}{z^2} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{i}{z^2} \xlongequal{\text{Step 2}} \frac{ 2 \cdot i }{ 1 \cdot z^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2i }{ z^2 } \end{aligned} $$

Subtract $ \dfrac{2i}{z^2} $ from $ \dfrac{3}{z} $ to get $ \dfrac{ \color{purple}{ -2iz+3z^2 } }{ z^3 }$.

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}{ z^2 }$ and the second by $\color{blue}{ z }$.

$$ \begin{aligned} \frac{3}{z} - \frac{2i}{z^2} & = \frac{ 3 \cdot \color{blue}{ z^2 }}{ z \cdot \color{blue}{ z^2 }} - \frac{ 2i \cdot \color{blue}{ z }}{ z^2 \cdot \color{blue}{ z }} = \\[1ex] &=\frac{ \color{purple}{ 3z^2 } }{ z^3 } - \frac{ \color{purple}{ 2iz } }{ z^3 }=\frac{ \color{purple}{ -2iz+3z^2 } }{ z^3 } \end{aligned} $$

Add $ \dfrac{-2iz+3z^2}{z^3} $ and $ \dfrac{-3z+2i}{z^2+1} $ to get $ \dfrac{ \color{purple}{ -2iz+3z^2 } }{ z^5+z^3 }$.

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}{ z^2+1 }$ and the second by $\color{blue}{ z^3 }$.

$$ \begin{aligned} \frac{-2iz+3z^2}{z^3} + \frac{-3z+2i}{z^2+1} & = \frac{ \left( -2iz+3z^2 \right) \cdot \color{blue}{ \left( z^2+1 \right) }}{ z^3 \cdot \color{blue}{ \left( z^2+1 \right) }} + \frac{ \left( -3z+2i \right) \cdot \color{blue}{ z^3 }}{ \left( z^2+1 \right) \cdot \color{blue}{ z^3 }} = \\[1ex] &=\frac{ \color{purple}{ -2iz^3-2iz+3z^4+3z^2 } }{ z^5+z^3 } + \frac{ \color{purple}{ -3z^4+2iz^3 } }{ z^5+z^3 } = \\[1ex] &=\frac{ \color{purple}{ -2iz+3z^2 } }{ z^5+z^3 } \end{aligned} $$