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+5i }\, $ is $ \color{blue}{ 2-5i } $.

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

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

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

$$ \begin{aligned} \left( 3-2i \right) \cdot \left( 2-5i \right) &= 3 \cdot 2 + 3 \cdot \left(-5 \,i \right) + \left( -2 \,i \right) \cdot \left(2 \right) + \left( -2 \,i \right) \cdot \left(-5 \,i \right) = \\[1 em] &= 6 -15 \, i -4 \, i + 10 \color{blue}{(-1)} = \\[1 em] &= -4-19i\end{aligned} $$ $$ \begin{aligned} \left( 2+5i \right) \cdot \left( 2-5i \right) &= 2 \cdot 2 + 2 \cdot \left(-5 \,i \right) + \left( 5 \,i \right) \cdot \left(2 \right) + \left( 5 \,i \right) \cdot \left(-5 \,i \right) = \\[1 em] &= 4 -10 \, i + 10 \, i -25 \color{blue}{(-1)} = \\[1 em] &= 29\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 3-2i }{ 2+5i } = \frac{ -4-19i }{ 29 } = \frac{ -4 }{ 29 } + \frac{ -19 }{ 29 } i= -\frac{ 4 }{ 29 }-\frac{ 19 }{ 29 }i $$

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