Step 1: Determine the conjugate of the denominator. ( to find the conjugate just change the sign of the imaginary part ).

In this example, the conjugate of $ \color{orangered}{ -4+2i }\, $ is $ \color{blue}{ -4-2i } $.

Step 2: Multiply both the numerator and denominator by the conjugate:

$$\begin{aligned} \frac{ 9i }{ -4+2i } &= \frac{ 9i }{ -4+2i } \cdot \frac{ \color{blue}{ -4-2i } }{ \color{blue}{ -4-2i } } = \\[1 em] &= \frac{ 9i \cdot \left( -4-2i \right) }{ \left( -4+2i \right) \cdot \left( -4-2i \right) } \end{aligned} $$

Step 3: Simplify numerator and denominator (use $\color{blue}{i^2 = -1}$)

$$ \begin{aligned} 9i \cdot \left( -4-2i \right) &= + \left( 9 \,i \right) \cdot \left(-4 \right) + \left( 9 \,i \right) \cdot \left(-2 \,i \right) = \\[1 em] &= -36 \, i -18 \color{blue}{(-1)} = \\[1 em] &= 18-36i\end{aligned} $$ $$ \begin{aligned} \left( -4+2i \right) \cdot \left( -4-2i \right) &= -4 \cdot \left(-4\right) -4 \cdot \left(-2 \,i \right) + \left( 2 \,i \right) \cdot \left(-4 \right) + \left( 2 \,i \right) \cdot \left(-2 \,i \right) = \\[1 em] &= 16 + 8 \, i -8 \, i -4 \color{blue}{(-1)} = \\[1 em] &= 20\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 9i }{ -4+2i } = \frac{ 18-36i }{ 20 } = \frac{ 18 }{ 20 } + \frac{ -36 }{ 20 } i= \frac{ 9 }{ 10 }-\frac{ 9 }{ 5 }i $$

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