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\left(x+1\right)a-2bx^{2}-4bx-2b
Consider ax+a-2bx^{2}-4bx-2b as a polynomial over variable a.
\left(x+1\right)\left(-2bx+a-2b\right)
Find one factor of the form ka+m, where ka divides the monomial with the highest power \left(x+1\right)a and m divides the constant factor -2bx^{2}-4bx-2b. One such factor is x+1. Factor the polynomial by dividing it by this factor.