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x^{2}+\left(4y-3\right)x+3y^{2}-7y+2
Consider x^{2}+4xy-3x+3y^{2}-7y+2 as a polynomial over variable x.
\left(x+y-2\right)\left(x+3y-1\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 3y^{2}-7y+2. One such factor is x+y-2. Factor the polynomial by dividing it by this factor.