Question

# Express as a polynomial. (3x+2)(x-5)(5x+4)

Polynomial factorization
Express as a polynomial.
$$\displaystyle{\left({3}{x}+{2}\right)}{\left({x}-{5}\right)}{\left({5}{x}+{4}\right)}$$

2021-02-12
Step 1
Given expression is
$$\displaystyle{\left({3}{x}+{2}\right)}{\left({x}-{5}\right)}{\left({5}{x}+{4}\right)}$$
It is known that
(a+b)(c+d)=a(c+d)+b(c+d)
Step 2
(3x+2)(x-5)(5x+4)=(3x+2)[(x-5)(5x+4)]
=(3x+2)[x(5x+4)-5(5x+4)]
$$\displaystyle={\left({3}{x}+{2}\right)}{\left[{5}{x}^{{{2}}}+{4}{x}-{25}{x}-{20}\right]}$$
$$\displaystyle={\left({3}{x}+{2}\right)}{\left[{5}{x}^{{{2}}}-{21}{x}-{20}\right]}$$
$$\displaystyle={3}{x}{\left[{5}{x}^{{{2}}}-{21}{x}-{20}\right]}+{2}{\left[{5}{x}^{{{2}}}-{21}{x}-{20}\right]}$$
$$\displaystyle={15}{x}^{{{3}}}-{63}{x}^{{{2}}}-{60}{x}+{10}{x}^{{{2}}}-{42}{x}-{40}$$
$$\displaystyle={15}{x}^{{{3}}}-{53}{x}^{{{2}}}-{102}{x}-{40}$$
Hence,
$$\displaystyle{\left({3}{x}+{2}\right)}{\left({x}-{5}\right)}{\left({5}{x}+{4}\right)}={15}{x}^{{{3}}}-{53}{x}^{{{2}}}-{102}{x}-{40}$$