To multiply two binomials, you may have heard the term "FOIL" for first, outside, inside, last.
$\displaylines{(x+2)(x-1) \\
x^2 -1x + 2x -2 \\
x^2 - 3x - 2}$
To multiply two polynomials where at least one has more than two terms, multiply each term in the first polynomial by each term in the second polynomial and combine like terms.
$\displaylines{(x^2 + x - 2)(x - 1) \\
x^3 + x^2 + x^2 + x - 2x - 2 \\
x^3 + 2x^2 - x -2}$
To multiply more than two polynomials, repeat the above process taking two polynomials at a time.
$\displaylines{(x+2)(x-1)(x+3) \\
(x^2 - 3x - 2)(x+3) \\
x^3 + 3x^2 - 3x^2 - 9x - 2x - 6 \\
x^3 - 11x -6}$
This same logic holds for two variables.
$\displaylines{(x+y)^3 \\
(x^2 + 2xy + y^2)(x + y) \\
x^3 + x^2y + 2x^2y + 2xy^2 + xy^2 + y^3 \\
x^3 + 3x^2y + 3xy^2 + y^3}$
Pascal's triangle can be used as a shortcut for exponentiation of $(x+y)$ to any power. To build Pascal's triangle, sum the neighboring terms (with 0 on the outsides). The result will be the coefficients of each term in the expanded polynomial.
$1 2 1$
$1 3 3 1$
$1 4 6 4 1$