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}{ 6+9i }\, $ is $ \color{blue}{ 6-9i } $.

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

$$\begin{aligned} \frac{ -6-2i }{ 6+9i } &= \frac{ -6-2i }{ 6+9i } \cdot \frac{ \color{blue}{ 6-9i } }{ \color{blue}{ 6-9i } } = \\[1 em] &= \frac{ \left( -6-2i \right) \cdot \left( 6-9i \right) }{ \left( 6+9i \right) \cdot \left( 6-9i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( -6-2i \right) \cdot \left( 6-9i \right) &= -6 \cdot 6 -6 \cdot \left(-9 \,i \right) + \left( -2 \,i \right) \cdot \left(6 \right) + \left( -2 \,i \right) \cdot \left(-9 \,i \right) = \\[1 em] &= -36 + 54 \, i -12 \, i + 18 \color{blue}{(-1)} = \\[1 em] &= -54+42i\end{aligned} $$ $$ \begin{aligned} \left( 6+9i \right) \cdot \left( 6-9i \right) &= 6 \cdot 6 + 6 \cdot \left(-9 \,i \right) + \left( 9 \,i \right) \cdot \left(6 \right) + \left( 9 \,i \right) \cdot \left(-9 \,i \right) = \\[1 em] &= 36 -54 \, i + 54 \, i -81 \color{blue}{(-1)} = \\[1 em] &= 117\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -6-2i }{ 6+9i } = \frac{ -54+42i }{ 117 } = \frac{ -54 }{ 117 } + \frac{ 42 }{ 117 } i= -\frac{ 6 }{ 13 }+\frac{ 14 }{ 39 }i $$

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