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

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

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

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

$$ \begin{aligned} \left( 1+8i \right) \cdot \left( -2+i \right) &= 1 \cdot \left(-2\right) + 1 \cdot \left(1 \,i \right) + \left( 8 \,i \right) \cdot \left(-2 \right) + \left( 8 \,i \right) \cdot \left(1 \,i \right) = \\[1 em] &= -2 + 1 \, i -16 \, i + 8 \color{blue}{(-1)} = \\[1 em] &= -10-15i\end{aligned} $$ $$ \begin{aligned} \left( -2-i \right) \cdot \left( -2+i \right) &= -2 \cdot \left(-2\right) -2 \cdot \left(1 \,i \right) + \left( -1 \,i \right) \cdot \left(-2 \right) + \left( -1 \,i \right) \cdot \left(1 \,i \right) = \\[1 em] &= 4 -2 \, i + 2 \, i -1 \color{blue}{(-1)} = \\[1 em] &= 5\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 1+8i }{ -2-i } = \frac{ -10-15i }{ 5 } = \frac{ -10 }{ 5 } + \frac{ -15 }{ 5 } i= -2-3i $$

Complex Division Calculator