Longer polynomials multiply the same way as binomials: every term of the first meets every term of the second. With more terms there are more products, so organization matters. Then combine like terms.
Why it works
The distributive property works for any number of terms: $$(x + 2)(x^2 - 3x + 4) = x(x^2 - 3x + 4) + 2(x^2 - 3x + 4)$$ A grid makes sure nothing is missed: one row per term of the first polynomial, one column per term of the second, and one product in each cell. A $2$-term polynomial times a $3$-term polynomial always gives $2 \times 3 = 6$ products before combining.
Key idea
Multiply every term by every term (count the products to be sure), then combine like terms and write the result in standard form.
Worked examples
Worked example: Binomial times trinomial
Multiply $(x + 2)(x^2 - 3x + 4)$.
- From $x$: $x^3 - 3x^2 + 4x$.
- From $2$: $2x^2 - 6x + 8$.
- Combine: $x^3 + (-3 + 2)x^2 + (4 - 6)x + 8$.
- Check with $x = 1$: $3 \cdot 2 = 6$ and $1 - 1 - 2 + 8 = 6$.
Answer: $x^3 - x^2 - 2x + 8$
Worked example: Trinomial times trinomial
Multiply $(x^2 + x - 1)(x^2 - x + 2)$.
- Nine products: $x^4 - x^3 + 2x^2$, then $x^3 - x^2 + 2x$, then $-x^2 + x - 2$.
- Combine: $x^4 + 0x^3 + 0x^2 + 3x - 2$.
Answer: $x^4 + 3x - 2$
Common mistake
Losing a term in the bookkeeping, or forgetting to combine like terms at the end. Count: an $m$-term times an $n$-term polynomial has $mn$ products. Then check by substituting $x = 1$ into both the original product and your answer.