Which of the following shows that polynomials form a closed system under multiplication? Option #1: 3(2x+5)=6x+15 Option #2: 8x+7=5x+3x+1+6 Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24 Option #4: 2x12⋅3=6x−−√ (1 point)

To determine which option shows that polynomials form a closed system under multiplication, we need to find a case where multiplying two polynomials yields another polynomial.

Let's analyze the options:

1. **Option #1: 3(2x+5)=6x+15**
This shows a polynomial multiplied by a constant (which is also a polynomial, but of degree 0). The result is a polynomial. However, it does not illustrate the multiplication of two polynomials of higher degree.

2. **Option #2: 8x+7=5x+3x+1+6**
This shows a sum of polynomials but does not demonstrate multiplication. It doesn’t address polynomial multiplication.

3. **Option #3: (2x²−4)(3y+6)=6x²y+12x²−12y−24**
This option shows the multiplication of two polynomials: \(2x² - 4\) and \(3y + 6\), resulting in another polynomial. This is an example of polynomial multiplication producing another polynomial.

4. **Option #4: 2x¹²⋅3=6x**
Here \(2x^{12}\) is a polynomial and 3 is a constant (also a polynomial of degree zero). The result is a polynomial, but like Option #1, it doesn't showcase the product of two non-constant polynomials.

Based on this analysis, **Option #3** is the correct choice because it demonstrates that multiplying two polynomials yields another polynomial, proving that polynomials form a closed system under multiplication.