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}{ 21-3i }\, $ is $ \color{blue}{ 21+3i } $.

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

$$\begin{aligned} \frac{ -4i }{ 21-3i } &= \frac{ -4i }{ 21-3i } \cdot \frac{ \color{blue}{ 21+3i } }{ \color{blue}{ 21+3i } } = \\[1 em] &= \frac{ \left( -4i \right) \cdot \left( 21+3i \right) }{ \left( 21-3i \right) \cdot \left( 21+3i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( -4i \right) \cdot \left( 21+3i \right) &= + \left( -4 \,i \right) \cdot \left(21 \right) + \left( -4 \,i \right) \cdot \left(3 \,i \right) = \\[1 em] &= -84 \, i -12 \color{blue}{(-1)} = \\[1 em] &= 12-84i\end{aligned} $$ $$ \begin{aligned} \left( 21-3i \right) \cdot \left( 21+3i \right) &= 21 \cdot 21 + 21 \cdot \left(3 \,i \right) + \left( -3 \,i \right) \cdot \left(21 \right) + \left( -3 \,i \right) \cdot \left(3 \,i \right) = \\[1 em] &= 441 + 63 \, i -63 \, i -9 \color{blue}{(-1)} = \\[1 em] &= 450\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ -4i }{ 21-3i } = \frac{ 12-84i }{ 450 } = \frac{ 12 }{ 450 } + \frac{ -84 }{ 450 } i= \frac{ 2 }{ 75 }-\frac{ 14 }{ 75 }i $$

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