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a^{2}-4a-b^{2}-2b+3
Consider a^{2}-b^{2}-4a-2b+3 as a polynomial over variable a.
\left(a+b-1\right)\left(a-b-3\right)
Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{2} and m divides the constant factor -b^{2}-2b+3. One such factor is a+b-1. Factor the polynomial by dividing it by this factor.