Question

Use the table to find each product.
(2x^2 + x - 3)(x^2 - 1)

2x^2 | x | -3 | —————
x^2 | answer | answer | answer |
-1 | answer | answer | answer |
——————————————
Combine like terms and find the final product
Product: ___________
options:
2x^4 + x^3 + 5x^2 - x + 3
2x^4 + x^3 - 5x^2 - x + 3

Options:
2x^4
-3x^2
-x
-2x^2
x^3
3

Answer 1

To find the product \((2x^2 + x - 3)(x^2 - 1)\), we can use the distributive property (also known as the FOIL method for binomials):

1. **Distributing \(2x^2\):**
\[
2x^2 \cdot x^2 = 2x^4
\]
\[
2x^2 \cdot -1 = -2x^2
\]

2. **Distributing \(x\):**
\[
x \cdot x^2 = x^3
\]
\[
x \cdot -1 = -x
\]

3. **Distributing \(-3\):**
\[
-3 \cdot x^2 = -3x^2
\]
\[
-3 \cdot -1 = 3
\]

Now, we collect all these products together:
\[
2x^4 + x^3 - 2x^2 - x - 3x^2 + 3
\]

Combine like terms:
- The \(x^2\) terms: \(-2x^2 - 3x^2 = -5x^2\)

Putting this all together gives us:
\[
2x^4 + x^3 - 5x^2 - x + 3
\]

Therefore, the final product is:
\[
\text{Product: } 2x^4 + x^3 - 5x^2 - x + 3
\]

This matches with the first option provided:
**Final answer:** **2x^4 + x^3 - 5x^2 - x + 3**