To determine which of the given responses shows that polynomials form a closed system under multiplication, we need to verify if multiplying the given expressions yields another polynomial.
Let's analyze each option:
1. **\( \frac{3x^2}{4} - 1 \)**:
- This expression itself is a polynomial (specifically, it's a quadratic polynomial). However, it does not demonstrate multiplication.
2. **\( \frac{1}{4}x^3 * 5 \)**:
- This simplifies to \( \frac{5}{4}x^3 \). This is still a polynomial (a cubic polynomial), and it shows multiplication of a polynomial by a constant. Therefore, it demonstrates closure under multiplication.
3. **\( (x)(\frac{1}{2} x) \)**:
- This multiplies to \( \frac{1}{2}x^2 \), which is a polynomial (a quadratic polynomial). This shows multiplication of polynomials resulting in another polynomial, hence demonstrating closure.
4. **\( (\sqrt{x})(6x * -3) \)**:
- The term \(6x * -3\) simplifies to \(-18x\), and multiplying this by \(\sqrt{x}\) gives \(-18x^{\frac{3}{2}}\). The expression \(-18x^{\frac{3}{2}}\) is not a polynomial because polynomials only have non-negative integer exponents.
Based on this analysis, the options that demonstrate polynomials are closed under multiplication are:
- **\( \frac{1}{4} x^3 * 5 \)**
- **\( (x)(\frac{1}{2} x) \)**
Thus, both **\( \frac{1}{4}x^3 * 5 \)** and **\( (x)(\frac{1}{2} x) \)** show that polynomials form a closed system under multiplication. If you need just one response, either of those would suffice; however, the clearest example showing closure is:
\[
\frac{1}{4}x^3 * 5
\]
This option shows multiplication leading to another polynomial directly.