1. **Problem statement:** (a) Factorize completely $12ab^2 + 18a^2b - 6b$ 2. **Formula and rules:** To factorize, find the greatest common factor (GCF) of all terms and factor it out. 3. **Step-by-step factorization:** - Identify GCF of coefficients: GCF of 12, 18, and 6 is 6. - Identify common variables: all terms have $b$, so factor out $b$. - So, GCF is $6b$. 4. **Factor out GCF:** $$12ab^2 + 18a^2b - 6b = 6b(\cancel{2} \times 2a b + \cancel{3} \times 3a^2 - \cancel{1})$$ More precisely: $$12ab^2 = 6b \times 2ab$$ $$18a^2b = 6b \times 3a^2$$ $$-6b = 6b \times (-1)$$ 5. **Write the factorized form:** $$12ab^2 + 18a^2b - 6b = 6b(2ab + 3a^2 - 1)$$ **Final answer:** $6b(2ab + 3a^2 - 1)$