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

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

$$\begin{aligned} \frac{ 6+2i }{ 3-i } &= \frac{ 6+2i }{ 3-i } \cdot \frac{ \color{blue}{ 3+i } }{ \color{blue}{ 3+i } } = \\[1 em] &= \frac{ \left( 6+2i \right) \cdot \left( 3+i \right) }{ \left( 3-i \right) \cdot \left( 3+i \right) } \end{aligned} $$

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

$$ \begin{aligned} \left( 6+2i \right) \cdot \left( 3+i \right) &= 6 \cdot 3 + 6 \cdot \left(1 \,i \right) + \left( 2 \,i \right) \cdot \left(3 \right) + \left( 2 \,i \right) \cdot \left(1 \,i \right) = \\[1 em] &= 18 + 6 \, i + 6 \, i + 2 \color{blue}{(-1)} = \\[1 em] &= 16+12i\end{aligned} $$ $$ \begin{aligned} \left( 3-i \right) \cdot \left( 3+i \right) &= 3 \cdot 3 + 3 \cdot \left(1 \,i \right) + \left( -1 \,i \right) \cdot \left(3 \right) + \left( -1 \,i \right) \cdot \left(1 \,i \right) = \\[1 em] &= 9 + 3 \, i -3 \, i -1 \color{blue}{(-1)} = \\[1 em] &= 10\end{aligned} $$

Step 4: Separate real and imaginary parts:

$$ \frac{ 6+2i }{ 3-i } = \frac{ 16+12i }{ 10 } = \frac{ 16 }{ 10 } + \frac{ 12 }{ 10 } i= \frac{ 8 }{ 5 }+\frac{ 6 }{ 5 }i $$

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