The polynomials:

\(\displaystyle{P}{\left({x}\right)}={8}{x}^{{4}}+{6}{x}^{{2}}-{3}{x}+{1},{D}{\left({x}\right)}={2}{x}^{{2}}-{x}+{2}\)

To determine:

The polynomials \(Q(x)\) and \(R(x)\) such that \(\displaystyle{P}{\left({x}\right)}={D}{\left({x}\right)}{Q}{\left({x}\right)}+{R}{\left({x}\right)}\)

Solution:

The long division of polynomial \(\displaystyle{P}{\left({x}\right)}={8}{x}^{{4}}+{6}{x}^{{2}}-{3}{x}+{1}\) by \(\displaystyle{D}{\left({x}\right)}={2}{x}^{{2}}-{x}+{2}\) is given by:

According to division algorithm, we have,

\(\displaystyle{8}{x}^{{4}}+{6}{x}^{{2}}-{3}{x}+{1}={\left({2}{x}^{{2}}-{x}+{2}\right)}{\left({4}{x}^{{2}}+{2}{x}\right)}+{\left(-{7}{x}+{1}\right)}\)

So, \(\displaystyle{Q}{\left({x}\right)}={4}{x}^{{2}}+{2}{x}\) and \(\displaystyle{R}{\left({x}\right)}=-{7}{x}+{1}\)

Conclusion:

Hence, \(\displaystyle{Q}{\left({x}\right)}={4}{x}^{{2}}+{2}{x}\) and \(\displaystyle{R}{\left({x}\right)}=-{7}{x}+{1}\)