To determine which of the given responses demonstrates that polynomials form a closed system under multiplication, we need to find an expression where the product of two polynomial expressions is also a polynomial.
**Analyzing each option:**
1. **\((x)(12x)\)**:
- The multiplication of \(x\) (a polynomial of degree 1) and \(12x\) (a polynomial of degree 1) results in \(12x^2\), which is also a polynomial.
- **This shows closure under multiplication.**
2. **\((\sqrt{x})(6x \cdot -3)\)**:
- \(\sqrt{x}\) is not a polynomial (it is a root) so this expression does not qualify as a polynomial multiplication and doesn't demonstrate closure under multiplication of polynomials.
3. **\(\frac{3x^2}{4} - 1\)**:
- This is a polynomial expression itself, but it is not a multiplication of two polynomials, so it does not demonstrate the closure property.
4. **\(14x^3 \cdot 5\)**:
- This multiplication results in \(70x^3\), which is a polynomial. However, because one of the factors is a constant (which is technically a polynomial of degree 0), this also shows closure.
**Conclusion:**
Both options 1 and 4 demonstrate closure under multiplication of polynomials, but option 1 (\((x)(12x)\)) is more straightforward as it consists purely of polynomials.
Thus, the response that clearly shows that polynomials form a closed system under multiplication is:
**\((x)(12x)\)**.