To multiply a binomial and a trinomial, you first set up a grid with the terms of each expression, then multiply corresponding terms and combine like terms. Specifically, this document provides steps to: 1) make a grid with the terms of the binomial (6x^2 + 3x) along one side and the trinomial (18x^3 + 7x^2 + 3x) along the other; 2) multiply the terms and combine like terms, resulting in the expression 54x^3 + 195x^2.