Add $a+bi$ and $ \dfrac{a-bi}{a^2+b^2} $ to get $ \dfrac{ \color{purple}{ a^2bi+b^3i+a^3+ab^2-bi+a } }{ a^2+b^2 }$.

Step 1: Write $ a+bi $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ a^2+b^2 }$.

$$ \begin{aligned} a+bi+ \frac{a-bi}{a^2+b^2} & \xlongequal{\text{Step 1}} \frac{a+bi}{\color{red}{1}} + \frac{a-bi}{a^2+b^2} = \frac{ \left( a+bi \right) \cdot \color{blue}{ \left( a^2+b^2 \right) }}{ 1 \cdot \color{blue}{ \left( a^2+b^2 \right) }} + \frac{ a-bi }{ a^2+b^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ a^3+ab^2+a^2bi+b^3i } }{ a^2+b^2 } + \frac{ \color{purple}{ a-bi } }{ a^2+b^2 } = \\[1ex] &=\frac{ \color{purple}{ a^2bi+b^3i+a^3+ab^2-bi+a } }{ a^2+b^2 } \end{aligned} $$