1. **State the problem:** Expand and simplify the expression $4u^3(2u^2 + 5u)$. 2. **Formula used:** Use the distributive property: $a(b + c) = ab + ac$. 3. **Apply distributive property:** Multiply $4u^3$ by each term inside the parentheses: $$4u^3 \times 2u^2 + 4u^3 \times 5u$$ 4. **Multiply coefficients and add exponents:** $$4 \times 2 = 8$$ $$u^3 \times u^2 = u^{3+2} = u^5$$ $$4 \times 5 = 20$$ $$u^3 \times u = u^{3+1} = u^4$$ 5. **Write the expanded expression:** $$8u^5 + 20u^4$$ 6. **Final answer:** $$\boxed{8u^5 + 20u^4}$$