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...