Dividing

A fraction with i in the bottom is not finished: the form $a+bi$ needs a real number underneath. To get one, multiply top and bottom by the conjugate of the bottom. That is allowed because you are multiplying by a fraction equal to 1.

Take $\frac{5+5i}{2+i}$. The conjugate of the bottom is $2-i$. The bottom becomes $\left(2+i\right)\left(2-i\right)$, which is 4 + 1, so 5. The top becomes $\left(5+5i\right)\left(2-i\right) = 10-5i+10i-5{i}^{2}$, which is $15+5i$. Then split the fraction: $\frac{15+5i}{5} = 3+i$.

Check by multiplying back: $\left(2+i\right)\left(3+i\right)$ is $6+5i+{i}^{2}$, which is $5+5i$.