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+15i }\, $ is $ \color{blue}{ -6-15i } $.

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

$$\begin{aligned} \frac{ 160+120i }{ -6+15i } &= \frac{ 160+120i }{ -6+15i } \cdot \frac{ \color{blue}{ -6-15i } }{ \color{blue}{ -6-15i } } = \\[1 em] &= \frac{ \left( 160+120i \right) \cdot \left( -6-15i \right) }{ \left( -6+15i \right) \cdot \left( -6-15i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 160+120i \right) \cdot \left( -6-15i \right) &= 160 \cdot \left(-6\right) + 160 \cdot \left(-15 \,i \right) + \left( 120 \,i \right) \cdot \left(-6 \right) + \left( 120 \,i \right) \cdot \left(-15 \,i \right) = \\[1 em] &= -960 -2400 \, i -720 \, i -1800 \color{blue}{(-1)} = \\[1 em] &= 840-3120i\end{aligned} $$ $$ \begin{aligned} \left( -6+15i \right) \cdot \left( -6-15i \right) &= -6 \cdot \left(-6\right) -6 \cdot \left(-15 \,i \right) + \left( 15 \,i \right) \cdot \left(-6 \right) + \left( 15 \,i \right) \cdot \left(-15 \,i \right) = \\[1 em] &= 36 + 90 \, i -90 \, i -225 \color{blue}{(-1)} = \\[1 em] &= 261\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 160+120i }{ -6+15i } = \frac{ 840-3120i }{ 261 } = \frac{ 840 }{ 261 } + \frac{ -3120 }{ 261 } i= \frac{ 280 }{ 87 }-\frac{ 1040 }{ 87 }i $$

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