(1)You write $$\dfrac{2}{3}b^5 - \dfrac{1}{6}b^3+\dfrac{4}{9}b^2 - 1=\dfrac{1}{18}\left((2b+1)(2b-1)\left(3b^3 + \dfrac{2(2b + 3)(2b-3)}{(2b+1)(2b-1)}\right)\right)$$ What happens when you substitute $b=\pm\dfrac{1}{2}$?

(2)The title of your threadFactoring polynomial,usually means to express a polynomial as a product of irreducible polynomial and does not include rational fractions. Otherwise, you should specify the terms of the factorization you are referring to.