$\begin{array}{l}
{x^4} - 3{x^3} + 4{x^2} - x - 3\\
= {x^2}\left( {{x^2} - 2x + 3} \right) - x\left( {{x^2} - 2x + 3} \right) - {x^2} + 2x - 3\\
= {x^2}\left( {{x^2} - 2x + 3} \right) - x\left( {{x^2} - 2x + 3} \right) - \left( {{x^2} - 2x + 3} \right)\\
= \left( {{x^2} - 2x + 3} \right)\left( {{x^2} - x - 1} \right)\\
\Rightarrow {x^4} - 3{x^3} + 4{x^2} - x - 3:\left( {{x^2} - 2x + 3} \right)\\
= {x^2} - x - 1
\end{array}$
