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x^{2}+\left(3y-4\right)x+2y^{2}-5y+3
Consider x^{2}+3xy+2y^{2}-4x-5y+3 as a polynomial over variable x.
\left(x+y-1\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 2y^{2}-5y+3. One such factor is x+y-1. Factor the polynomial by dividing it by this factor.