Step 1

we have to find all the rational zeros of the polynomial and write the polynomial in factored form.

\(\displaystyle{P}{\left({x}\right)}={2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}\)

Step 2

\(\displaystyle{2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}\)

\(\displaystyle{\left({2}{x}-{1}\right)}{\left({x}^{{{2}}}+{4}{x}+{4}\right)}\)

\(\displaystyle{\left({2}{x}-{1}\right)}{\left({x}+{2}\right)}^{{{2}}}{\left({2}{x}-{1}\right)}{\left({x}+{2}\right)}{\left({x}+{2}\right)}\)

therefore

\(\displaystyle{2}{x}^{{{3}}}+{7}{x}^{{{2}}}+{4}{x}-{4}={\left({2}{x}-{1}\right)}{\left({x}+{2}\right)}{\left({x}+{2}\right)}\)

now as we have to find the rational roots

therefore

\((2x-1)(x+2)(x+2)=0\)

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