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}{ 7+2i }\, $ is $ \color{blue}{ 7-2i } $.

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

$$\begin{aligned} \frac{ 5+3i }{ 7+2i } &= \frac{ 5+3i }{ 7+2i } \cdot \frac{ \color{blue}{ 7-2i } }{ \color{blue}{ 7-2i } } = \\[1 em] &= \frac{ \left( 5+3i \right) \cdot \left( 7-2i \right) }{ \left( 7+2i \right) \cdot \left( 7-2i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 5+3i \right) \cdot \left( 7-2i \right) &= 5 \cdot 7 + 5 \cdot \left(-2 \,i \right) + \left( 3 \,i \right) \cdot \left(7 \right) + \left( 3 \,i \right) \cdot \left(-2 \,i \right) = \\[1 em] &= 35 -10 \, i + 21 \, i -6 \color{blue}{(-1)} = \\[1 em] &= 41+11i\end{aligned} $$ $$ \begin{aligned} \left( 7+2i \right) \cdot \left( 7-2i \right) &= 7 \cdot 7 + 7 \cdot \left(-2 \,i \right) + \left( 2 \,i \right) \cdot \left(7 \right) + \left( 2 \,i \right) \cdot \left(-2 \,i \right) = \\[1 em] &= 49 -14 \, i + 14 \, i -4 \color{blue}{(-1)} = \\[1 em] &= 53\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 5+3i }{ 7+2i } = \frac{ 41+11i }{ 53 } = \frac{ 41 }{ 53 } + \frac{ 11 }{ 53 } i= \frac{ 41 }{ 53 }+\frac{ 11 }{ 53 }i $$

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