Use the following multiplication problem to show how multiplying polynomials is a closed system. (x^2−1)(−2x+5) (2 points)

To demonstrate that multiplying polynomials is a closed system using the multiplication problem \((x^2 - 1)(-2x + 5)\), we need to perform the multiplication and show that the result is also a polynomial.

### Step 1: Distribute the terms

We will use the distributive property (also known as the FOIL method for binomials) to multiply the two polynomials:

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

Distributing \(x^2\) over \(-2x + 5\):

\[
x^2 \cdot -2x = -2x^3
\]
\[
x^2 \cdot 5 = 5x^2
\]

Now distributing \(-1\) over \(-2x + 5\):

\[
-1 \cdot -2x = 2x
\]
\[
-1 \cdot 5 = -5
\]

### Step 2: Combine the results

Now we combine all the results from the distributions:

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

### Final Result

The final result of multiplying the two polynomials \( (x^2 - 1)(-2x + 5) \) is:

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

### Conclusion

The result \(-2x^3 + 5x^2 + 2x - 5\) is a polynomial, demonstrating that the multiplication of two polynomials results in another polynomial. This property is what makes the set of polynomials a closed system under multiplication. Thus, we can conclude that multiplying polynomials is indeed a closed system.