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