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------------------------------------------------ What is the product?

[tex]\[

\left(7x^2\right)\left(2x^3+5\right)\left(x^2-4x-9\right)

\][/tex]

A. [tex]$14x^5 - x^4 - 46x^3 - 58x^2 - 20x - 45$[/tex]

B. [tex]$14x^6 - 56x^5 - 91x^4 - 140x^3 - 315x^2$[/tex]

C. [tex]$14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2$[/tex]

D. [tex]$14x^{12} - 182x^6 + 35x^4 - 455x^2$[/tex]

To find the product of the given expression [tex]$(7x^2)(2x^3 + 5)(x^2 - 4x - 9)$[/tex], we need to multiply these three expressions together. I'll walk you through the steps to arrive at the final product.

### Step 1: Multiply the first two expressions

First, we'll multiply [tex]$7x^2$[/tex] by [tex]$(2x^3 + 5)$[/tex].

- Distribute [tex]$7x^2$[/tex] to each term in [tex]$2x^3 + 5$[/tex]:

[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]

[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]

Adding these results together gives:

[tex]\[
14x^5 + 35x^2
\][/tex]

### Step 2: Multiply the result with the third expression

Now, we need to multiply [tex]$(14x^5 + 35x^2)$[/tex] by [tex]$(x^2 - 4x - 9)$[/tex].

Distribute each term in [tex]$(14x^5 + 35x^2)$[/tex] across every term in [tex]$(x^2 - 4x - 9)$[/tex]:

1. Distribute [tex]$14x^5$[/tex]:
- [tex]$14x^5 \cdot x^2 = 14x^7$[/tex]
- [tex]$14x^5 \cdot (-4x) = -56x^6$[/tex]
- [tex]$14x^5 \cdot (-9) = -126x^5$[/tex]

2. Distribute [tex]$35x^2$[/tex]:
- [tex]$35x^2 \cdot x^2 = 35x^4$[/tex]
- [tex]$35x^2 \cdot (-4x) = -140x^3$[/tex]
- [tex]$35x^2 \cdot (-9) = -315x^2$[/tex]

### Step 3: Combine all the terms

Now, combine all like terms from these results:

[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This is the expanded form and the product of the original expression [tex]$(7x^2)(2x^3 + 5)(x^2 - 4x - 9)$[/tex].