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

To demonstrate that polynomials form a closed system under multiplication, we need to show that the product of any two polynomials is also a polynomial.

Let's analyze each of the options provided:

1. **(x^2)(1x)**:
- This is the product of two polynomials (x^2 is a polynomial and 1x is also a polynomial). Multiplying them gives \(x^{2+1} = x^3\), which is a polynomial.

2. **(x^2 + 1)(x - 12)**:
- This is the product of two polynomials (x^2 + 1 and x - 12). When we apply the distributive property (FOIL), we get:
\[
(x^2)(x) + (x^2)(-12) + (1)(x) + (1)(-12) = x^3 - 12x^2 + x - 12
\]
This is a polynomial.

3. **(x−√)(x + 1)**:
- This expression seems to imply some form of operations that involve irrational numbers rather than being purely polynomial terms (it looks unusual and may not be interpreted correctly as written). However, if we assume we meant (x - √a)(x + 1) where a is a constant, the product could include irrational elements, making it more complex to determine closure over the integers or rationals. If none of its roots produce a non-polynomial term, it might still yield a polynomial.

4. **x^2 + 2x + 1**:
- This is a polynomial itself, but it doesn't demonstrate multiplication of two polynomials.

**Conclusion**: The second option, **(x^2 + 1)(x - 12)**, clearly shows the multiplication of two polynomials yielding another polynomial, confirming that polynomials are closed under multiplication. Option 1 also confirms closure. Option 3 could be ambiguous based on its representation, and option 4 is not a product of two polynomials.

Therefore, the responses that demonstrate closure under multiplication are **(x^2)(1x)** and **(x^2 + 1)(x - 12)**.