How many nonzero terms are in the expansion of $$(x+4)(2x^2+3x+9)-3(x^3-2x^2+7x)?$$
I want to simplify this expression by multiplying out the brackets and combining like terms. To multiply out the first bracket, I can use the distributive property: $$(x+4)(2x^2+3x+9) = x(2x^2+3x+9) + 4(2x^2+3x+9).$$ Then I can multiply each term by the corresponding coefficient: $$x(2x^2+3x+9) + 4(2x^2+3x+9) = 2x^3+3x^2+9x + 8x^2+12x+36.$$ I can combine the like terms to get: $$2x^3+3x^2+9x + 8x^2+12x+36 = 2x^3+11x^2+21x+36.$$ To subtract the second bracket, I can use the distributive property again: $$2x^3+11x^2+21x+36 - 3(x^3-2x^2+7x) = 2x^3+11x^2+21x+36 - 3x^3 + 6x^2 - 21x.$$ I can combine the like terms again to get: $$2x^3+11x^2+21x+36 - 3x^3 + 6x^2 - 21x = -x^3 + 17x^2 + 36.$$ This is the simplified expression, and I can see that it has three nonzero terms. 