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+5i }\, $ is $ \color{blue}{ 4-5i } $.

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

$$\begin{aligned} \frac{ 2+3i }{ 4+5i } &= \frac{ 2+3i }{ 4+5i } \cdot \frac{ \color{blue}{ 4-5i } }{ \color{blue}{ 4-5i } } = \\[1 em] &= \frac{ \left( 2+3i \right) \cdot \left( 4-5i \right) }{ \left( 4+5i \right) \cdot \left( 4-5i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 2+3i \right) \cdot \left( 4-5i \right) &= 2 \cdot 4 + 2 \cdot \left(-5 \,i \right) + \left( 3 \,i \right) \cdot \left(4 \right) + \left( 3 \,i \right) \cdot \left(-5 \,i \right) = \\[1 em] &= 8 -10 \, i + 12 \, i -15 \color{blue}{(-1)} = \\[1 em] &= 23+2i\end{aligned} $$ $$ \begin{aligned} \left( 4+5i \right) \cdot \left( 4-5i \right) &= 4 \cdot 4 + 4 \cdot \left(-5 \,i \right) + \left( 5 \,i \right) \cdot \left(4 \right) + \left( 5 \,i \right) \cdot \left(-5 \,i \right) = \\[1 em] &= 16 -20 \, i + 20 \, i -25 \color{blue}{(-1)} = \\[1 em] &= 41\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 2+3i }{ 4+5i } = \frac{ 23+2i }{ 41 } = \frac{ 23 }{ 41 } + \frac{ 2 }{ 41 } i= \frac{ 23 }{ 41 }+\frac{ 2 }{ 41 }i $$

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