$$ 3i^3 = 3 \cdot \color{blue}{i^2} \cdot i = 3 \cdot ( \color{blue}{-1}) \cdot i = -3 \cdot \, i $$

$$ -9i^2 = -9 \cdot (-1) = 9 $$

Combine like terms:

$$ \color{blue}{-3i} + \color{red}{9} \color{blue}{-6i} + \color{red}{4} = \color{blue}{-9i} + \color{red}{13} $$

Add $-9i+13$ and $ \dfrac{2}{i} $ to get $ \dfrac{ \color{purple}{ -9i^2+13i+2 } }{ i }$.

Step 1: Write $ -9i+13 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} -9i+13+ \frac{2}{i} & \xlongequal{\text{Step 1}} \frac{-9i+13}{\color{red}{1}} + \frac{2}{i} = \frac{ \left( -9i+13 \right) \cdot \color{blue}{ i }}{ 1 \cdot \color{blue}{ i }} + \frac{ 2 }{ i } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -9i^2+13i } }{ i } + \frac{ \color{purple}{ 2 } }{ i }=\frac{ \color{purple}{ -9i^2+13i+2 } }{ i } \end{aligned} $$

$$ -9i^2 = -9 \cdot (-1) = 9 $$

Simplify numerator

$$ \color{blue}{9} +13i+ \color{blue}{2} = 13i+ \color{blue}{11} $$

Divide $ \, 11+13i \, $ by $ \, i \, $ to get $\,\, 13-11i $.
( view steps )

Combine like terms:

$$ -11i+13 = -11i+13 $$