The quadratic formula used for quadratic equation \(\displaystyle{a}{x}^{{2}}+{b}{x}+{c}={0}\) is,

\(\displaystyle{x}=-\frac{{{b}\pm\sqrt{{{b}^{{2}}-{4}{a}{c}}}}}{{{2}{a}}}\)

Apply the quadratic formula in \(\displaystyle{3}{x}^{{2}}+{5}{x}+{2}={0}\) then

\(\displaystyle{x}=\frac{{-{5}\pm\sqrt{{{5}^{{2}}-{4}\times{3}\times{2}}}}}{{{2}\times{3}}}\)

\(\displaystyle{x}={\left(-{5}\pm\frac{\sqrt{{{25}-{24}}}}{{6}}\right)}\)

\(\displaystyle{x}=\frac{{-{5}\pm{1}}}{{6}}\)

\(\displaystyle{x}=-\frac{{2}}{{3}},-{1}\)