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