When multiplying polynomials of higher degree than quadradics (cubics * cubics or cubics * quartics, for example) instead of writing out each term using the distributive property, you can make a simple 2 x 2 matrix with the coefficients of one equation in a row and the other in a column. You are still using the distributive property multiplying each term by every other term, but visually it is easier to keep track of like terms because you can sum them up in diagonals rather than picking them out amongst a long string of numbers and letters.

For example: (2x^3 + x^2 - 4x + 1) * (x^4 + 5x^3 + 2x - 5) =

2 1 -4 1 Diagonals start at 2 go to down to -10, then over to -5

------------------

1 | 2 1 -4 1

5 | 10 5 -20 5

0 | 0 0 0 0

2 | 4 2 -8 2

-5|-10 -5 20 -5

Adding diagonals (summing all terms lower left to upper right) gives you:

2x^7 + 11x^6 + x^5 - 15x^4 - 3x^3 - 13x^2 + 22x - 5

Note that the x^2 term is missing in the second equation, so we must put 0's in line three. It should be clear that using the traditional method would have resulted in 16 terms laid out horizontally (4 x 4) so this method makes it much simpler and less prone to errors because additions and subtractions are all close to each other visually.