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}{ -2+6i }\, $ is $ \color{blue}{ -2-6i } $.

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

$$\begin{aligned} \frac{ 4+5i }{ -2+6i } &= \frac{ 4+5i }{ -2+6i } \cdot \frac{ \color{blue}{ -2-6i } }{ \color{blue}{ -2-6i } } = \\[1 em] &= \frac{ \left( 4+5i \right) \cdot \left( -2-6i \right) }{ \left( -2+6i \right) \cdot \left( -2-6i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 4+5i \right) \cdot \left( -2-6i \right) &= 4 \cdot \left(-2\right) + 4 \cdot \left(-6 \,i \right) + \left( 5 \,i \right) \cdot \left(-2 \right) + \left( 5 \,i \right) \cdot \left(-6 \,i \right) = \\[1 em] &= -8 -24 \, i -10 \, i -30 \color{blue}{(-1)} = \\[1 em] &= 22-34i\end{aligned} $$ $$ \begin{aligned} \left( -2+6i \right) \cdot \left( -2-6i \right) &= -2 \cdot \left(-2\right) -2 \cdot \left(-6 \,i \right) + \left( 6 \,i \right) \cdot \left(-2 \right) + \left( 6 \,i \right) \cdot \left(-6 \,i \right) = \\[1 em] &= 4 + 12 \, i -12 \, i -36 \color{blue}{(-1)} = \\[1 em] &= 40\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 4+5i }{ -2+6i } = \frac{ 22-34i }{ 40 } = \frac{ 22 }{ 40 } + \frac{ -34 }{ 40 } i= \frac{ 11 }{ 20 }-\frac{ 17 }{ 20 }i $$

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