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x^{2}+\left(5y+5\right)x-6y^{2}-12y-6
Consider x^{2}+5xy+5x-6y^{2}-12y-6 as a polynomial over variable x.
\left(x+6y+6\right)\left(x-y-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 -6y^{2}-12y-6. One such factor is x+6y+6. Factor the polynomial by dividing it by this factor.