Multiplying Polynomials by Monomial
To multiply polynomial by monomial, one should use distributive property of multiplication.
Then, just multiply monomials and you're done.
Example 1. Multiply $$${\left({2}{x}\right)}{\left({3}{{x}}^{{2}}+{5}{x}+{4}\right)}$$$.
$$${\color{red}{{{\left({2}{x}\right)}}}}{\color{green}{{{\left({3}{{x}}^{{2}}+{5}{x}+{4}\right)}}}}=$$$
$$$={\color{red}{{{\left({2}{x}\right)}}}}\cdot{\color{green}{{{\left({3}{{x}}^{{2}}\right)}}}}+{\color{red}{{{\left({2}{x}\right)}}}}\cdot{\color{green}{{{\left({5}{x}\right)}}}}+{\color{red}{{{\left({2}{x}\right)}}}}\cdot{\color{green}{{{\left({4}\right)}}}}=$$$ (distributive property of multiplication)
$$$={6}{{x}}^{{3}}+{10}{{x}}^{{2}}+{8}{x}$$$ (multiply monomials)
Answer: $$${\left({2}{x}\right)}{\left({3}{{x}}^{{2}}+{5}{x}+{4}\right)}={6}{{x}}^{{3}}+{10}{{x}}^{{2}}+{8}{x}$$$.
Negative terms are handled in the same way.
Example 2. Multiply the following: $$${\left({{x}}^{{3}}-{5}{{x}}^{{2}}-{x}+{7}\right)}\frac{{1}}{{3}}{{x}}^{{2}}$$$.
$$${\color{green}{{{\left({{x}}^{{3}}-{5}{{x}}^{{2}}-{x}+{7}\right)}}}}{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}=$$$
$$$={\color{green}{{{{x}}^{{3}}}}}\cdot{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}+{\color{green}{{{\left(-{5}{{x}}^{{2}}\right)}}}}\cdot{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}+{\color{green}{{{\left(-{x}\right)}}}}\cdot{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}+{\color{green}{{{7}}}}\cdot{\color{red}{{\frac{{1}}{{3}}{{x}}^{{2}}}}}=$$$ (distributive property of multiplication)
$$$=\frac{{1}}{{3}}{{x}}^{{5}}-\frac{{5}}{{3}}{{x}}^{{4}}-\frac{{1}}{{3}}{{x}}^{{3}}+\frac{{7}}{{3}}{{x}}^{{2}}$$$ (multiply monomials)
Answer: $$${\left({{x}}^{{3}}-{5}{{x}}^{{2}}-{x}\right)}\frac{{1}}{{3}}{{x}}^{{2}}=\frac{{1}}{{3}}{{x}}^{{5}}-\frac{{5}}{{3}}{{x}}^{{4}}-\frac{{1}}{{3}}{{x}}^{{3}}+\frac{{7}}{{3}}{{x}}^{{2}}$$$.
Of course, polynomials with many variables can also be handled in a similar way.
Example 3. Multiply $$$-{3}{x}{{y}}^{{2}}$$$ by $$${\left({3}{{x}}^{{2}}{y}+{2}{x}{z}-{5}{x}{{y}}^{{2}}-{z}\right)}$$$.
$$$-{3}{x}{{y}}^{{2}}{\left({3}{{x}}^{{2}}{y}+{2}{x}{z}-{5}{x}{{y}}^{{2}}-{z}\right)}=$$$
$$$={\left(-{3}{x}{{y}}^{{2}}\right)}{\left({3}{{x}}^{{2}}{y}\right)}+{\left(-{3}{x}{{y}}^{{2}}\right)}{\left({2}{x}{z}\right)}+{\left(-{3}{x}{{y}}^{{2}}\right)}{\left(-{5}{x}{{y}}^{{2}}\right)}+{\left(-{3}{x}{{y}}^{{2}}\right)}{\left(-{z}\right)}=$$$
$$$=-{9}{{x}}^{{3}}{{y}}^{{3}}-{6}{{x}}^{{2}}{{y}}^{{2}}{z}+{15}{{x}}^{{2}}{{y}}^{{4}}+{3}{x}{{y}}^{{2}}{z}$$$.
Answer: $$$-{3}{x}{{y}}^{{2}}{\left({3}{{x}}^{{2}}{y}+{2}{x}{z}-{5}{x}{{y}}^{{2}}-{z}\right)}=-{9}{{x}}^{{3}}{{y}}^{{3}}-{6}{{x}}^{{2}}{{y}}^{{2}}{z}+{15}{{x}}^{{2}}{{y}}^{{4}}+{3}{x}{{y}}^{{2}}{z}$$$.
Now, it is time to exercise.
Exercise 1. Multiply $$${\left({{x}}^{{3}}+{2}{x}+{4}\right)}\cdot{\left({5}{{x}}^{{2}}\right)}$$$.
Answer: $$${5}{{x}}^{{5}}+{10}{{x}}^{{3}}+{20}{{x}}^{{2}}$$$.
Exercise 2. Multiply $$$-\frac{{2}}{{7}}{{a}}^{{3}}{\left({{a}}^{{3}}-{2}{{a}}^{{2}}+{7}{b}\right)}$$$.
Answer: $$$-\frac{{2}}{{7}}{{a}}^{{6}}+\frac{{4}}{{7}}{{a}}^{{5}}-{2}{{a}}^{{3}}{b}$$$.
Exercise 3. Multiply $$${\left(-{3}{a}{b}\right)}{\left({5}{{a}}^{{2}}{b}-{3}{{a}}^{{3}}{b}{c}+\frac{{3}}{{5}}{{a}}^{{2}}{{b}}^{{2}}-\frac{{1}}{{10}}{a}{b}\right)}$$$.
Answer: $$$-{15}{{a}}^{{3}}{{b}}^{{2}}+{9}{{a}}^{{4}}{{b}}^{{2}}{c}-\frac{{9}}{{5}}{{a}}^{{3}}{{b}}^{{3}}+\frac{{3}}{{10}}{{a}}^{{2}}{{b}}^{{2}}$$$.