How do you multiple and divide complex numbers?

A complex number is a binomial . . . . ( a + ib ).

To multiply two of them, just FOIL them, like you would  (2x+1)(x-3) .
But while you're FOILing around, just be careful how you handle the " i "s.

To divide them is another story.  I took a whole semester course
in complex numbers once, and I think we spent about 2 weeks
on dividing them.  Your question is valid, but I think the answer is
too complicated to try and present here ... and plus I don't remember
how to do it ... so I'm simply going to not answer the second part
but take your points anyway.  



I just read the other answer, posted by  Andrewmj2002, and
he has some good stuff there for dividing.  I just want to clean up
some of what he said:

-- To divide complex numbers, you'll make a fraction, with one
complex number in the numerator, and the other one in the
denominator.

-- Find the complex conjugate of the denominator.  That's just
a new complex number with the same real part but the negative
of the same imaginary part.

Example:  The complex conjugate of  (A + iC)  is  (A - iC) .

--  Multiply both the numerator and denominator of your big fraction
by this conjugate of the denominator.
Since you're multiplying both the numerator and denominator by
the same quantity, you're not changing the value of the fraction. 
But after you finish FOILing the numerator and denominator,
the numerator becomes a single, complex binomial, and the
denominator becomes all real !

--  Simplify, and you have the quotient of your original complex numbers.