1. **State the problem:** Factor the expression $4ab^2 - 36a$ completely. 2. **Identify the greatest common factor (GCF):** Both terms have a factor of $4a$ because $4$ divides both $4$ and $36$, and $a$ is common in both terms. 3. **Factor out the GCF:** $$4ab^2 - 36a = 4a(b^2 - 9)$$ 4. **Recognize the difference of squares:** $$b^2 - 9 = b^2 - 3^2$$ 5. **Apply the difference of squares formula:** $$x^2 - y^2 = (x - y)(x + y)$$ 6. **Factor the difference of squares:** $$b^2 - 9 = (b - 3)(b + 3)$$ 7. **Write the fully factored form:** $$4a(b - 3)(b + 3)$$ **Final answer:** $4a(b - 3)(b + 3)$