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

To determine whether polynomials form a closed system under multiplication, we need to verify if the product of any two polynomials is also a polynomial. Let's analyze each of the responses you provided:

1. **1/4x^2 × 5**:
- This can be simplified to (1/4) * 5 * x^2 = (5/4)x^2, which is a polynomial.

2. **(x) (1/2x)**:
- This simplifies to (1/2)x^2, which is a polynomial.

3. **3x^2/4 - 1**:
- This expression represents a polynomial. It is not a multiplication statement but still represents a polynomial.

4. **(x squared) (6x × -3)**:
- This can be rewritten as \(x^2 \cdot (-18x) = -18x^3\), which is a polynomial.

All of the responses either show polynomial multiplication or represent polynomials, therefore confirming that multiplying polynomials yields polynomials, thereby demonstrating that polynomials are closed under multiplication.

If we are specifically looking for responses that directly illustrate multiplication leading to a polynomial, then responses 1, 2, and 4 are the best examples. However, all the above responses support the conclusion that polynomials form a closed system under multiplication.