1. **State the problem:** Identify the error in generating an expression equivalent to $$4 + 2x - \frac{1}{2} (10 - 4x)$$ and then correct the error. 2. **Original expression:** $$4 + 2x - \frac{1}{2} (10 - 4x)$$ 3. **Distribute the $$-\frac{1}{2}$$ across the parentheses:** $$4 + 2x - \frac{1}{2} \times 10 + \frac{1}{2} \times 4x$$ 4. **Calculate each term:** $$4 + 2x - 5 + 2x$$ 5. **Combine like terms:** $$4 - 5 + 2x + 2x = (4 - 5) + (2x + 2x) = -1 + 4x$$ 6. **Error in the original work:** The original simplification incorrectly canceled $$2x$$ terms, resulting in $$4 - 5 = -1$$, omitting the $$4x$$ term. 7. **Correct final expression:** $$-1 + 4x$$ **Summary:** The error was in incorrectly canceling $$2x$$ terms. The correct simplification keeps the $$4x$$ term.