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x^{2}+x+4y-16y^{2}
Consider x+4y+x^{2}-16y^{2} as a polynomial over variable x.
\left(x+4y\right)\left(x-4y+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 -16y^{2}+4y. One such factor is x+4y. Factor the polynomial by dividing it by this factor.