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x^{2}+\left(2y+2\right)x+y^{2}+2y-15
Consider x^{2}+y^{2}+2xy+2x+2y-15 as a polynomial over variable x.
\left(x+y-3\right)\left(x+y+5\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 y^{2}+2y-15. One such factor is x+y-3. Factor the polynomial by dividing it by this factor.