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}{ -6-8i }\, $ is $ \color{blue}{ -6+8i } $.

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

$$\begin{aligned} \frac{ -9+5i }{ -6-8i } &= \frac{ -9+5i }{ -6-8i } \cdot \frac{ \color{blue}{ -6+8i } }{ \color{blue}{ -6+8i } } = \\[1 em] &= \frac{ \left( -9+5i \right) \cdot \left( -6+8i \right) }{ \left( -6-8i \right) \cdot \left( -6+8i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( -9+5i \right) \cdot \left( -6+8i \right) &= -9 \cdot \left(-6\right) -9 \cdot \left(8 \,i \right) + \left( 5 \,i \right) \cdot \left(-6 \right) + \left( 5 \,i \right) \cdot \left(8 \,i \right) = \\[1 em] &= 54 -72 \, i -30 \, i + 40 \color{blue}{(-1)} = \\[1 em] &= 14-102i\end{aligned} $$ $$ \begin{aligned} \left( -6-8i \right) \cdot \left( -6+8i \right) &= -6 \cdot \left(-6\right) -6 \cdot \left(8 \,i \right) + \left( -8 \,i \right) \cdot \left(-6 \right) + \left( -8 \,i \right) \cdot \left(8 \,i \right) = \\[1 em] &= 36 -48 \, i + 48 \, i -64 \color{blue}{(-1)} = \\[1 em] &= 100\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -9+5i }{ -6-8i } = \frac{ 14-102i }{ 100 } = \frac{ 14 }{ 100 } + \frac{ -102 }{ 100 } i= \frac{ 7 }{ 50 }-\frac{ 51 }{ 50 }i $$

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