1. **Problem Statement:** Add the polynomials $2x^2 - 5xy + y^3$ and $4x + y$. 2. **Formula and Rules:** To add polynomials, combine like terms. Like terms have the same variables raised to the same powers. 3. **Step-by-step Solution:** - Write the polynomials: $$2x^2 - 5xy + y^3 + 4x + y$$ - Group like terms: $$2x^2 + (-5xy) + y^3 + 4x + y$$ - There are no like terms to combine because each term has different variables or powers. 4. **Final Answer:** $$2x^2 - 5xy + y^3 + 4x + y$$ This is the sum of the polynomials.