If u is odd and w is even, what is `(uw)^(2)+u`?

Answer

To solve the problem, we need to analyze the expression \((uw)^2 + u\) given that \(u\) is an odd number and \(w\) is an even number.

### Step-by-Step Solution:

1. **Identify the nature of \(u\) and \(w\)**:
- \(u\) is odd.
- \(w\) is even.

2. **Calculate the product \(uw\)**:
- When an odd number is multiplied by an even number, the result is always even.
- Therefore, \(uw\) is even.

3. **Square the product \(uw\)**:
- Now, we need to find \((uw)^2\).
- Since \(uw\) is even, squaring an even number results in another even number.
- Thus, \((uw)^2\) is even.

4. **Add \(u\) to \((uw)^2\)**:
- We have \((uw)^2 + u\).
- We established that \((uw)^2\) is even and \(u\) is odd.
- The sum of an even number and an odd number is always odd.

5. **Conclusion**:
- Therefore, \((uw)^2 + u\) is odd.

### Final Answer:
\((uw)^2 + u\) is odd.