Let’s first perform the long division. Just remember that we keep going until the remainder has degree that is strictly less that the degree of the polynomial we’re dividing by, \({x^2} - 3x + 1\) in this case. The polynomial we’re dividing by has degree two and so, in this case, we’ll stop when the remainder is degree one or zero.

Here is the long division work for this problem.

\[\require{enclose}\begin{align*} & 2{{x}^{3}}+7{{x}^{2}}+19x+50 \\ {{x}^{2}}-3x+1 & {\enclose{longdiv}{{2{{x}^{5}}+{{x}^{4}}-6x+9}}} \\ - & \underline{\left( 2{{x}^{5}}-6{{x}^{4}}+2{{x}^{3}} \right)} \\ & \hspace{0.75in}7{{x}^{4}}-2{{x}^{3}}-6x+9 \\ & \hspace{0.5in}-\underline{\left( 7{{x}^{4}}-21{{x}^{3}}+7{{x}^{2}} \right)} \\ & \hspace{1.4in}19{{x}^{3}}-7{{x}^{2}}-6x+9 \\ & \hspace{1.05in}-\underline{\left( 19{{x}^{3}}-57{{x}^{2}}+19x \right)} \\ & \hspace{2.0in}50{{x}^{2}}-25x+9 \\ & \hspace{1.75in}-\underline{\left( 50{{x}^{2}}-150x+50 \right)} \\ & \hspace{2.75in}125x-41 \\ \end{align*}\] Show Step 2