In the article on complex numbers it was written that the calculation
rules for real numbers should also apply to complex numbers.
This article describes how to multiplicate complex numbers.

As an example we use the two numbers \(3 + i\) and \(1 - 2i\).

So it should be calculated \((3+i)·(1-2i)\)

According to the principle of permanence, the calculation rules of real numbers should continue to apply.
We will therefore first multiply the parenthesis as normal. So we write

Additional to expressions with \(i\) in the formula appears \(i^2\).
We can easily replace this \(i^2\).
By definition \(i\) is \(i^2 = -1\).
So we replace \(i^2\) with the number \(-1\) and continue to calculate with the result from above as usual.

\(3-6i+i-2i^2=3-6i+i-2·(-1)=3-5i+2=5-5i\)

The result of the calculation is \(5 - 5i\).

This article describes the multiplication of complex numbers in normal form.
Easier to calculate is the Multiplication of complex numbers in polar form.