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x^{2}-3x-4y^{2}-6y
Consider x^{2}-4y^{2}-3x-6y as a polynomial over variable x.
\left(x+2y\right)\left(x-2y-3\right)
Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{2} and m divides the constant factor -4y^{2}-6y. One such factor is x+2y. Factor the polynomial by dividing it by this factor.