### Theory:

In this multiplication, we have to multiply each of the terms in the first polynomial by each of the terms in the second polynomial.
Important!
First, we should look for like terms, if any, and combine them before starting multiplication of polynomial by a polynomial.
Example:
Consider a trinomial $$a+b-c$$ and a polynomial $$2a-3b+5c$$.

Let us multiply the trinomial $$a+b-c$$ by a polynomial $$2a-3b+5c$$.

Note we have to multiply each of the terms in the first polynomial $$a+b-c$$ by each of the terms in the second polynomial $$2a-3b+5c$$.

$\left(a+b-c\right)×\left(2a-3b+5c\right)$

$=\left(a×2a\right)+\left(a×-3b\right)+\left(a×5c\right)+\left(b×2a\right)+\left(b×-3b\right)+\left(b×5c\right)+\left(-c×2a\right)+\left(-c×-3b\right)+\left(-c×5c\right)$

$=2{a}^{2}-3\mathit{ab}+5\mathit{ac}+2\mathit{ab}-3{b}^{2}+5\mathit{bc}-2\mathit{ac}+3\mathit{bc}-5{c}^{2}$

$=2{a}^{2}-3{b}^{2}-5{c}^{2}-\mathit{ab}+8\mathit{bc}+3\mathit{ac}$.