There are several methods for multiplying polynomials.

One method is to use the property of distribution of multiplication over addition.  This method can be used to multiply a polynomial by a monomial, or a polynomial with another polynomial, also know as repeated distribution or horizontal multiplication. 

Polynomials can also be multiplied vertically and by using a table.

When multiplying two binomials, a special case of repeated distribution may be used, called FOIL.

These class notes provide further explanation and examples.