Which of the following responses demonstrates that polynomials form a closed system under multiplication?

To determine which of the responses demonstrates that polynomials form a closed system under multiplication, we need to see if the results of the multiplications presented yield a polynomial.

A polynomial is an expression that can be written in the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \) where \( a_i \) are constants (coefficients) and \( n \) is a non-negative integer.

Let's analyze each option:

A) \( (x^2 + 1)(x - \frac{1}{2}) \)

This is the product of two polynomials. The result will be:
\[
x^2 \cdot x - \frac{1}{2} x^2 + 1 \cdot x - \frac{1}{2} = x^3 - \frac{1}{2}x^2 + x - \frac{1}{2}
\]
This expression is a polynomial.

B) \( (x)(x + 1) \)

This is also the product of two polynomials. The result will be:
\[
x^2 + x
\]
This expression is a polynomial.

C) \( x^2 + 2x + 1 \)

This expression is already given as a polynomial and does not involve a multiplication operation.

D) \( \left(\frac{x^2}{2}\right) \left(\frac{1}{x}\right) \)

This is the product of two expressions, but \( \frac{1}{x} \) is not a polynomial. The result would be:
\[
\frac{x^2}{2} \cdot \frac{1}{x} = \frac{x}{2}
\]
While \( \frac{x}{2} \) is a polynomial, if we consider that \( \frac{1}{x} \) is not a polynomial, this does not demonstrate closure under multiplication for polynomials.

From the options provided, both A and B demonstrate that polynomials form a closed system under multiplication, as their results are polynomials. However, since the question likely expects only one response:

**Final Answer:** A) \((x^2 + 1)(x - \frac{1}{2})\) or B) \((x)(x + 1)\) (both demonstrate closure). If forced to select one: A.