Step 1:

Distribute each term of the first polynomial to every term of the second polynomial.

\(\displaystyle{\left({2}{x}^{{{2}}}\ +\ {4}{y}^{{{3}}}\right)}\ {\left({3}{x}^{{{3}}}\ +\ {4}{y}^{{{2}}}\right)}={2}{x}^{{{2}}}\ {\left({3}{x}^{{{3}}}\right)}\ +\ {2}{x}^{{{2}}}\ {\left({4}{y}^{{{2}}}\right)}\ +\ {4}{y}^{{{3}}}\ {\left({3}{x}^{{{3}}}\right)}\ +\ {4}{y}^{{{3}}}\)

\(\displaystyle={6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}\)

\(\displaystyle+\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}\)

Step 2: Combine like terms. In this problem, there are no line terms.

\(\displaystyle{6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}\ +\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}\)

Conclusion:

Polynomial with Polynomial: To multiply a polynomial and a polynomial, use the distributive property until every term of one polynomial is mutiplied times every term of the other polynomial. Make sure that you simplify your answer by combining any like terms.

Example: \(\displaystyle{\left({2}{x}^{{{2}}}\ +\ {4}{y}^{{{3}}}\right)}\ {\left({3}{x}^{{{3}}}\ +\ {4}{y}^{{{2}}}\right)}={6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}\ +\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}.\)

Distribute each term of the first polynomial to every term of the second polynomial.

\(\displaystyle{\left({2}{x}^{{{2}}}\ +\ {4}{y}^{{{3}}}\right)}\ {\left({3}{x}^{{{3}}}\ +\ {4}{y}^{{{2}}}\right)}={2}{x}^{{{2}}}\ {\left({3}{x}^{{{3}}}\right)}\ +\ {2}{x}^{{{2}}}\ {\left({4}{y}^{{{2}}}\right)}\ +\ {4}{y}^{{{3}}}\ {\left({3}{x}^{{{3}}}\right)}\ +\ {4}{y}^{{{3}}}\)

\(\displaystyle={6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}\)

\(\displaystyle+\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}\)

Step 2: Combine like terms. In this problem, there are no line terms.

\(\displaystyle{6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}\ +\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}\)

Conclusion:

Polynomial with Polynomial: To multiply a polynomial and a polynomial, use the distributive property until every term of one polynomial is mutiplied times every term of the other polynomial. Make sure that you simplify your answer by combining any like terms.

Example: \(\displaystyle{\left({2}{x}^{{{2}}}\ +\ {4}{y}^{{{3}}}\right)}\ {\left({3}{x}^{{{3}}}\ +\ {4}{y}^{{{2}}}\right)}={6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}\ +\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}.\)