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]

Let's find the product of the polynomial expression [tex]$(7x^2)(2x^3 + 5)(x^2 - 4x - 9)$[/tex].

### Step 1: Multiply [tex]$7x^2$[/tex] with [tex]$(2x^3 + 5)$[/tex]

First, distribute [tex]$7x^2$[/tex] across the terms in the binomial:

- Multiply [tex]$7x^2$[/tex] and [tex]$2x^3$[/tex]:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]

- Multiply [tex]$7x^2$[/tex] and [tex]$5$[/tex]:
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]

So, [tex]$(7x^2)(2x^3 + 5) = 14x^5 + 35x^2$[/tex].

### Step 2: Multiply [tex]$(14x^5 + 35x^2)$[/tex] with [tex]$(x^2 - 4x - 9)$[/tex]

Now distribute each term in [tex]$(14x^5 + 35x^2)$[/tex] across each term in [tex]$(x^2 - 4x - 9)$[/tex].

#### Multiply [tex]$14x^5$[/tex] with [tex]$(x^2 - 4x - 9)$[/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]

#### Multiply [tex]$35x^2$[/tex] with [tex]$(x^2 - 4x - 9)$[/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, let's write out all the terms we have from our multiplication:

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

Make sure to combine terms with the same powers of [tex]$x$[/tex], if any. In this problem, each term has a unique power of [tex]$x$[/tex], so there's no simplification needed.

So, the expanded product of the expression is:

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

This is the final answer.