Complex numbers are of the form a + ib, with a real component a and an imaginary component which has a coefficient b.
When you multiply two real numbers it is equivalent to multiplying 4 numbers. Multiply each of a and ib of the complex number with both the a and ib of the other number. Once you are done, the terms with i can be added to give the coefficient of the imaginary component of the product. Use the property that i^2 = -1 to deal with the term that does not have i and which has i^2.
We'll explain the multiplication of complex numbers written in algebraic form, choosing 2 comeplx numbers:
z1 = a + bi and z2 = c + di
The real part of z1 = Re(z1) = a
The imaginary part of z1 = Im(z1) = b
The real part of z2 = Re(z2) = c
The imaginary part of z2 = Im(z2) = d
We'll multiply the numbers z1*z2 = (a+bi)(c+di)
We'll apply FOIL:
z1*z2 = a*c + a*di + c*bi + b*d*i^2
Rule number one: i^2 = -1
z1*z2 = a*c + a*di + c*bi - b*d
Rule number 2: We'll combine real parts and imaginary parts:
z1*z2 = (ac-bd) + i(ad+bc)