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}{ -4+i }\, $ is $ \color{blue}{ -4-i } $.

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

$$\begin{aligned} \frac{ 8-2i }{ -4+i } &= \frac{ 8-2i }{ -4+i } \cdot \frac{ \color{blue}{ -4-i } }{ \color{blue}{ -4-i } } = \\[1 em] &= \frac{ \left( 8-2i \right) \cdot \left( -4-i \right) }{ \left( -4+i \right) \cdot \left( -4-i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 8-2i \right) \cdot \left( -4-i \right) &= 8 \cdot \left(-4\right) + 8 \cdot \left(-1 \,i \right) + \left( -2 \,i \right) \cdot \left(-4 \right) + \left( -2 \,i \right) \cdot \left(-1 \,i \right) = \\[1 em] &= -32 -8 \, i + 8 \, i + 2 \color{blue}{(-1)} = \\[1 em] &= -34\end{aligned} $$ $$ \begin{aligned} \left( -4+i \right) \cdot \left( -4-i \right) &= -4 \cdot \left(-4\right) -4 \cdot \left(-1 \,i \right) + \left( 1 \,i \right) \cdot \left(-4 \right) + \left( 1 \,i \right) \cdot \left(-1 \,i \right) = \\[1 em] &= 16 + 4 \, i -4 \, i -1 \color{blue}{(-1)} = \\[1 em] &= 17\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 8-2i }{ -4+i } = \frac{ -34 }{ 17 } = \frac{ -34 }{ 17 } + \frac{ 0 }{ 17 } i= -2 $$

Complex Division Calculator