1. **Stating the problem:** Expand and simplify the expression $2x(x + 4) + x(2x + 7)$. 2. **Using the distributive law:** Multiply each term inside the parentheses by the term outside. $$2x(x + 4) = 2x \cdot x + 2x \cdot 4 = 2x^2 + 8x$$ $$x(2x + 7) = x \cdot 2x + x \cdot 7 = 2x^2 + 7x$$ 3. **Combine the expanded terms:** $$2x^2 + 8x + 2x^2 + 7x$$ 4. **Group like terms:** $$ (2x^2 + 2x^2) + (8x + 7x)$$ 5. **Add the coefficients:** $$4x^2 + 15x$$ 6. **Final simplified expression:** $$\boxed{4x^2 + 15x}$$ This is the fully expanded and simplified form of the original expression.