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}{ -5+6i }\, $ is $ \color{blue}{ -5-6i } $.

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

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

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

$$ \begin{aligned} \left( -6+i \right) \cdot \left( -5-6i \right) &= -6 \cdot \left(-5\right) -6 \cdot \left(-6 \,i \right) + \left( 1 \,i \right) \cdot \left(-5 \right) + \left( 1 \,i \right) \cdot \left(-6 \,i \right) = \\[1 em] &= 30 + 36 \, i -5 \, i -6 \color{blue}{(-1)} = \\[1 em] &= 36+31i\end{aligned} $$ $$ \begin{aligned} \left( -5+6i \right) \cdot \left( -5-6i \right) &= -5 \cdot \left(-5\right) -5 \cdot \left(-6 \,i \right) + \left( 6 \,i \right) \cdot \left(-5 \right) + \left( 6 \,i \right) \cdot \left(-6 \,i \right) = \\[1 em] &= 25 + 30 \, i -30 \, i -36 \color{blue}{(-1)} = \\[1 em] &= 61\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -6+i }{ -5+6i } = \frac{ 36+31i }{ 61 } = \frac{ 36 }{ 61 } + \frac{ 31 }{ 61 } i= \frac{ 36 }{ 61 }+\frac{ 31 }{ 61 }i $$

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