Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials

The Closure Property states that the product of two polynomials is also a polynomial. To demonstrate this using multiplication, let's take two simple polynomials and multiply them together.

Consider the following two polynomials:

1. \( P(x) = 2x + 3 \)
2. \( Q(x) = x^2 + 1 \)

Now we will multiply these two polynomials:

\[
P(x) \cdot Q(x) = (2x + 3)(x^2 + 1)
\]

Using the distributive property (also known as the FOIL method in the case of binomials), we can calculate the product:

\[
= 2x \cdot x^2 + 2x \cdot 1 + 3 \cdot x^2 + 3 \cdot 1
\]

Calculating each term gives:

\[
= 2x^3 + 2x + 3x^2 + 3
\]

Now, let's rearrange the terms in standard form:

\[
= 2x^3 + 3x^2 + 2x + 3
\]

The result \( 2x^3 + 3x^2 + 2x + 3 \) is itself a polynomial.

Thus, this example demonstrates the Closure Property of multiplication of polynomials: multiplying two polynomials \( P(x) \) and \( Q(x) \) results in another polynomial.