Factor \(x^3-3x^2+x-3\text{.}\) To succeed with this example, we will need to “factor out” a trivial number \(1\) that isn't apparent until we make it so.

\begin{align*} x^3-3x^2+x-3\amp=\left(x^3-3x^2\right)+(x-3)\\ \amp=\highlight{x^2}(x-3)\highlight{{}+1}(x-3)\\ \amp=\highlight{x^2}\attention{\overbrace{(x-3)}}\highlight{{}+1}\attention{\overbrace{(x-3)}}\\ \amp=(x-3)\highlight{\left(x^2+1\right)} \end{align*}

Notice how we changed \(x-3\) to \(+1(x-3)\text{,}\) so we wouldn't forget the \(+1\) in the final factored form. As always, we should check this is correct by multiplying it out.