Factorize $a^2-ab-bc\pm c^2$ I got this question in a test but it did not specify the variable with respect to which I was supposed to factorize $$a^2-ab-bc\pm c^2$$ where it could be just $a(a-b)-c(b\pm c)$ but no common factor over all terms. I feel I may be missing something. The $\pm$ is there because I cannot remember whether the last sign was minus or plus. **Is there some trick to factorize this or is this question vacuous? What does it mean to factorize this?**

We observe that

\begin{eqnarray} a^2-ab-bc+c^2&=&a^2+c^2-b(a+c)\\\ &=&(a+c)^2-b(a+c)-2ac\\\ &=&(a+c)^2-b(a+c)+\frac{b^2}{4}-\frac{b^2+8ac}{4}\\\ &=&\left(a+c-\frac{b}{2}\right)^2-\frac{b^2+8ac}{4}. \end{eqnarray} Hence, if $b^2+8ac\geq 0$ then $$ a^2-ab-bc+c^2=\left(a+c-\frac{b}{2}+\sqrt{\frac{b^2+8ac}{4}}\right)\left(a+c-\frac{b}{2}-\sqrt{\frac{b^2+8ac}{4}}\right) $$