Use the following multiplication problem to show how multiplying polynomials is a closed system.

To show that multiplying polynomials is a closed system, we can take the given polynomials \((x^2−1)\) and \((-2x+5)\) and perform the multiplication.

### Step 1: Distribute the terms

We apply the distributive property (also known as the FOIL method for binomials).

\[
(x^2 - 1)(-2x + 5)
\]

We can distribute each term in the first polynomial to each term in the second polynomial:

1. Multiply \(x^2\) by \(-2x\):
\[
x^2 \cdot -2x = -2x^3
\]

2. Multiply \(x^2\) by \(5\):
\[
x^2 \cdot 5 = 5x^2
\]

3. Multiply \(-1\) by \(-2x\):
\[
-1 \cdot -2x = 2x
\]

4. Multiply \(-1\) by \(5\):
\[
-1 \cdot 5 = -5
\]

### Step 2: Combine the results

Now, we combine all the terms together:

\[
-2x^3 + 5x^2 + 2x - 5
\]

### Step 3: Arrange the expression

We can arrange the final polynomial in standard form, which typically orders the terms from highest to lowest degree:

\[
-2x^3 + 5x^2 + 2x - 5
\]

### Conclusion

The result, \(-2x^3 + 5x^2 + 2x - 5\), is also a polynomial. This shows that the product of two polynomials (both of degree 2 and 1, respectively) is still a polynomial (in this case, degree 3).

Therefore, the multiplication of polynomials is a closed system, as the product of two polynomials is also a polynomial.

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