To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of two polynomials is also a polynomial.
Let's analyze the given options:
1. **\((x^2/2)(1/x)\)**:
- This simplifies to \((x^2/2)(1/x) = x/2\), which is not a polynomial because it contains a term with a negative exponent (the \(1/x\) contributes a power of \(-1\)). Thus, this does not demonstrate closure.
2. **\(x^2 + 2x + 1\)**:
- This is just a polynomial by itself. It does not show multiplication.
3. **\((x^2 + 1)(x - \frac{1}{2})\)**:
- When you multiply these two polynomials together, you will get another polynomial (since multiplying polynomials results in a polynomial). This is a valid response that demonstrates closure.
4. **\((x - \sqrt{})(x + 1)\)**:
- Depending on what the terms inside the square root are, if they are not fully specified, we can't evaluate this properly. However, if this represents multiplying two expressions with square roots (not particularly polynomial terms), it may break polynomial form depending on the complexity of the square root.
The best answer that clearly shows that polynomials form a closed system under multiplication is:
\[
\boxed{(x^2 + 1)(x - \frac{1}{2})}
\]
This response demonstrates that multiplying two polynomials results in another polynomial, satisfying the condition for closure under multiplication.