To determine which response demonstrates that polynomials form a closed system under multiplication, we need to remember that a set is closed under an operation if performing that operation on members of the set always produces a member of the same set.
Let's analyze each option:
1. **\(x^2 + 2x + 1 \cdot x\)**:
- This simplifies to \(x^3 + 2x^2 + x\), which is a polynomial. Thus, it shows closure under multiplication.
2. **\((x^2/2)(1/x)\)**:
- This simplifies to \(\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2}\), which is not a polynomial since it has a rational coefficient and can be expressed as a non-integral power of \(x\) (specifically, \(x^{-1}\)). This does not demonstrate closure under multiplication.
3. **\((x^2 + 1)(x - 1/2)\)**:
- This is a product of two polynomials and will result in another polynomial. Thus, it shows closure under multiplication.
4. **\((x - \sqrt{x})(x + 1)\)**:
- While both \(x - \sqrt{x}\) and \(x + 1\) look like polynomials, the term \(\sqrt{x}\) is not a polynomial, as it can be expressed as \(x^{1/2}\), which is a non-integer exponent. Therefore, this does not represent a polynomial.
**Conclusion**:
The options that demonstrate polynomials form a closed system under multiplication are:
1. \(x^2 + 2x + 1 \cdot x\)
2. \((x^2 + 1)(x - 1/2)\)
If we need to choose one, the first option is a straightforward case of polynomial multiplication:
\[
\boxed{x^2 + 2x + 1 \cdot x}
\]
This confirms polynomials' closure under multiplication effectively.