Everything times everything
Multiplying polynomials rests on one idea: the distributive property, applied as many times as needed. Every term of the first polynomial must multiply every term of the second. When you multiply two terms, multiply the coefficients and add the exponents — that is the product rule of exponents: x^2 · x^3 = x^5.
The simplest case is a monomial times a polynomial: distribute the single term across each piece inside the parentheses. This is the same expanding move you have done since you first multiplied 3(x + 2).
2x(3x^2 − 5x + 4) = 2x·3x^2 + 2x·(−5x) + 2x·4 = 6x^3 − 10x^2 + 8x
FOIL: two binomials
When you multiply two binomials, there are exactly four products. FOIL is a mnemonic for not missing any: First terms, Outer terms, Inner terms, Last terms. FOIL is not a new rule — it is just the distributive property bookkeeping for the special two-by-two case.
(x + 3)(x − 5) F: x·x = x^2 O: x·(−5) = −5x I: 3·x = 3x L: 3·(−5) = −15 = x^2 − 5x + 3x − 15 = x^2 − 2x − 15 combine the like middle terms
Bigger products, same idea
For a binomial times a trinomial, distribute each term of the binomial over the whole trinomial, then combine like terms. Work in standard form and line up matching powers as you go — it makes the final tidy-up almost automatic.
(x + 2)(x^2 − 3x + 4)
x·(x^2 − 3x + 4) = x^3 − 3x^2 + 4x
2·(x^2 − 3x + 4) = 2x^2 − 6x + 8
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= x^3 − x^2 − 2x + 8