To determine which statement shows how the product of [tex]\((x+3)^2\)[/tex] demonstrates the closure property of multiplication, let's follow the steps of expanding and simplifying the expression [tex]\((x+3)^2\)[/tex].
1. First, expand [tex]\((x+3)^2\)[/tex]:
[tex]\[
(x + 3)^2 = (x + 3)(x + 3)
\][/tex]
2. Apply the distributive property:
[tex]\[
(x + 3)(x + 3) = x(x + 3) + 3(x + 3)
\][/tex]
3. Distribute [tex]\(x\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[
x(x + 3) + 3(x + 3) = x^2 + 3x + 3x + 9
\][/tex]
4. Combine like terms:
[tex]\[
x^2 + 3x + 3x + 9 = x^2 + 6x + 9
\][/tex]
Now we have the expression [tex]\(x^2 + 6x + 9\)[/tex]. This expression is a polynomial because it is a sum of terms, each of which is a product of a constant and a non-negative integer power of [tex]\(x\)[/tex].
Therefore, the correct statement is:
[tex]\[ x^2 + 6x + 9 \text{ is a polynomial} \][/tex]
This shows that the product of [tex]\((x+3)^2\)[/tex] is a polynomial, which demonstrates the closure property of multiplication in this context. The specific answer to the question is:
[tex]\[ x^2 + 6x + 9 \text{ is a polynomial} \][/tex]
