1. **State the problem:** Factorise the expression $$4b^4 + 6b^3 + 2b^2$$. 2. **Identify common factors:** Look for the greatest common factor (GCF) in all terms. Each term contains a factor of $$2b^2$$. 3. **Extract the GCF:** Factor out $$2b^2$$ from each term: $$4b^4 + 6b^3 + 2b^2 = 2b^2(2b^2) + 2b^2(3b) + 2b^2(1) = 2b^2(2b^2 + 3b + 1)$$ 4. **Factor the quadratic inside the parentheses:** Factor $$2b^2 + 3b + 1$$. 5. **Use the AC method:** Multiply $$2 \times 1 = 2$$. Find two numbers that multiply to 2 and add to 3: these are 1 and 2. 6. **Rewrite the middle term:** $$2b^2 + 3b + 1 = 2b^2 + 2b + b + 1$$ 7. **Group terms:** $$(2b^2 + 2b) + (b + 1)$$ 8. **Factor each group:** $$2b(b + 1) + 1(b + 1)$$ 9. **Factor out the common binomial:** $$(2b + 1)(b + 1)$$ 10. **Write the fully factorised form:** $$4b^4 + 6b^3 + 2b^2 = 2b^2(2b + 1)(b + 1)$$ **Final answer:** $$\boxed{2b^2(2b + 1)(b + 1)}$$