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}{ 4150-244i }\, $ is $ \color{blue}{ 4150+244i } $.

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

$$\begin{aligned} \frac{ 4150 }{ 4150-244i } &= \frac{ 4150 }{ 4150-244i } \cdot \frac{ \color{blue}{ 4150+244i } }{ \color{blue}{ 4150+244i } } = \\[1 em] &= \frac{ 4150 \cdot \left( 4150+244i \right) }{ \left( 4150-244i \right) \cdot \left( 4150+244i \right) } \end{aligned} $$

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

$$ \begin{aligned} 4150 \cdot \left( 4150+244i \right) &= 4150 \cdot 4150 + 4150 \cdot \left(244 \,i \right) = \\[1 em] &= 17222500 + 1012600 \, i = \\[1 em] &= 17222500+1012600i\end{aligned} $$ $$ \begin{aligned} \left( 4150-244i \right) \cdot \left( 4150+244i \right) &= 4150 \cdot 4150 + 4150 \cdot \left(244 \,i \right) + \left( -244 \,i \right) \cdot \left(4150 \right) + \left( -244 \,i \right) \cdot \left(244 \,i \right) = \\[1 em] &= 17222500 + 1012600 \, i -1012600 \, i -59536 \color{blue}{(-1)} = \\[1 em] &= 17282036\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 4150 }{ 4150-244i } = \frac{ 17222500+1012600i }{ 17282036 } = \frac{ 17222500 }{ 17282036 } + \frac{ 1012600 }{ 17282036 } i= \frac{ 4305625 }{ 4320509 }+\frac{ 253150 }{ 4320509 }i $$

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