1. **State the problem:** Simplify the sum of the two polynomials: $$(2u^3 + 6u^2 + 3) + (2u^3 - 7u + 6)$$ 2. **Formula and rules:** To add polynomials, combine like terms. Like terms have the same variable raised to the same power. 3. **Identify like terms:** - $2u^3$ and $2u^3$ are like terms. - $6u^2$ has no like term in the second polynomial. - $-7u$ has no like term in the first polynomial. - Constants $3$ and $6$ are like terms. 4. **Add like terms:** - $2u^3 + 2u^3 = 4u^3$ - $6u^2$ remains as is. - $-7u$ remains as is. - $3 + 6 = 9$ 5. **Write the simplified polynomial:** $$4u^3 + 6u^2 - 7u + 9$$ 6. **Conclusion:** The sum simplifies to $4u^3 + 6u^2 - 7u + 9$. This matches option 4 in the multiple-choice question.