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:

$$\begin{aligned} \frac{ -11-27i }{ 4-i } &= \frac{ -11-27i }{ 4-i } \cdot \frac{ \color{blue}{ 4+i } }{ \color{blue}{ 4+i } } = \\[1 em] &= \frac{ \left( -11-27i \right) \cdot \left( 4+i \right) }{ \left( 4-i \right) \cdot \left( 4+i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( -11-27i \right) \cdot \left( 4+i \right) &= -11 \cdot 4 -11 \cdot \left(1 \,i \right) + \left( -27 \,i \right) \cdot \left(4 \right) + \left( -27 \,i \right) \cdot \left(1 \,i \right) = \\[1 em] &= -44 -11 \, i -108 \, i -27 \color{blue}{(-1)} = \\[1 em] &= -17-119i\end{aligned} $$ $$ \begin{aligned} \left( 4-i \right) \cdot \left( 4+i \right) &= 4 \cdot 4 + 4 \cdot \left(1 \,i \right) + \left( -1 \,i \right) \cdot \left(4 \right) + \left( -1 \,i \right) \cdot \left(1 \,i \right) = \\[1 em] &= 16 + 4 \, i -4 \, i -1 \color{blue}{(-1)} = \\[1 em] &= 17\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -11-27i }{ 4-i } = \frac{ -17-119i }{ 17 } = \frac{ -17 }{ 17 } + \frac{ -119 }{ 17 } i= -1-7i $$

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