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x^{2}+\left(4y+4\right)x-12y^{2}-24y-12
Consider x^{2}+4xy+4x-12y^{2}-24y-12 as a polynomial over variable x.
\left(x+6y+6\right)\left(x-2y-2\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 -12y^{2}-24y-12. One such factor is x+6y+6. Factor the polynomial by dividing it by this factor.