You may want to review multiplying polynomials by monomials first.

Multiplying polynomials can take a long time. Suppose you have
to multiply *x*^{3} − *x*^{2}
− 1 by *x*^{4} − *x* + 2. Then you
need to multiply *each* term of the first one
by *each* term of the second. Since in this case our two
polynomials each have three terms, you'll have to do a total of 3
* 3 = 9 multiplications. To keep the signs straight, I recommend
keeping the minus signs with the terms. Here we go.

*x*^{3}**x*^{4}=*x*^{7}.*x*^{3}* (−*x*) = −*x*^{4}.*x*^{3}* 2 = 2*x*^{3}.- −
*x*^{2}**x*^{4}= −*x*^{6}. - −
*x*^{2}* (−*x*) =*x*^{3}. - −
*x*^{2}* 2 = −2*x*^{2}. - −1 *
*x*^{4}= −*x*^{4}. - −1 * (−
*x*) =*x*. - −1 * 2 = −2.

Now, we put it all together. Remember that, just like with adding
polynomials, we put a plus sign if there's no minus sign. So we
get *x*^{7} − *x*^{4} +
2*x*^{3} − *x*^{6}
+ *x*^{3} − 2*x*^{2}
− *x*^{4} + *x* − 2. Finally, we
collect like terms, so our answer is *x*^{7} −
2*x*^{4} + 3*x*^{3}
− *x*^{6} − 2*x*^{2}
+ *x* − 2. (We could rearrange this so the exponents
would be in descending order, but you don't have to.)