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a^{2}+\left(2b+1\right)a+b^{2}+b
Consider a^{2}+2ab+b^{2}+a+b as a polynomial over variable a.
\left(a+b\right)\left(a+b+1\right)
Find one factor of the form a^{k}+m, where a^{k} divides the monomial with the highest power a^{2} and m divides the constant factor b^{2}+b. One such factor is a+b. Factor the polynomial by dividing it by this factor.