1. **State the problem:** Factorise the expression $$4b^4 + 6b^3 + 2b^2$$. 2. **Identify common factors:** Each term contains a factor of $$b^2$$, so we can factor that out first. 3. **Factor out $$b^2$$:** $$4b^4 + 6b^3 + 2b^2 = b^2(4b^2 + 6b + 2)$$ 4. **Factor the quadratic inside the parentheses:** Look for common factors in $$4b^2 + 6b + 2$$. Each coefficient is divisible by 2. 5. **Factor out 2:** $$b^2(4b^2 + 6b + 2) = b^2 \times 2(2b^2 + 3b + 1) = 2b^2(2b^2 + 3b + 1)$$ 6. **Check if $$2b^2 + 3b + 1$$ can be factored further:** Find two numbers that multiply to $$2 \times 1 = 2$$ and add to $$3$$. These are $$1$$ and $$2$$. 7. **Factor the quadratic:** $$2b^2 + 3b + 1 = (2b + 1)(b + 1)$$ 8. **Write the fully factored form:** $$4b^4 + 6b^3 + 2b^2 = 2b^2(2b + 1)(b + 1)$$ **Final answer:** $$\boxed{2b^2(2b + 1)(b + 1)}$$