1. State the problem: Factorize the expression $16ab^3 - 8ab + 32a^2bc^2$. 2. Identify the common factors of all terms: Each term contains at least $8ab$. 3. Factor out $8ab$: $$16ab^3 - 8ab + 32a^2bc^2 = 8ab(2b^2 - 1 + 4ac^2)$$ 4. Check the expression inside the parentheses for further factorization: $2b^2 - 1 + 4ac^2$ cannot be factored further using simple methods. 5. Therefore, the fully factorized form is: $$8ab(2b^2 - 1 + 4ac^2)$$