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]

Sure! Let's solve the problem by multiplying the expressions step-by-step:

We need to find the product of [tex]$(7x^2)(2x^3+5)(x^2-4x-9)$[/tex].

### Step 1: Expand the first pair of terms

First, multiply [tex]$7x^2$[/tex] with the polynomial [tex]$2x^3 + 5$[/tex]:

[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]

Calculate each term:

- [tex]$7x^2 \cdot 2x^3 = 14x^{5}$[/tex]
- [tex]$7x^2 \cdot 5 = 35x^2$[/tex]

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

### Step 2: Multiply the result with the remaining polynomial

Now, take the result [tex]$14x^5 + 35x^2$[/tex] and multiply it with [tex]$x^2 - 4x - 9$[/tex]:

[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]

Distribute each term from the first polynomial through the second polynomial:

- [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]

- [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 like terms

Combine all the terms we found:

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

This is the expanded and simplified product of the original expression.

Therefore, the final product is:

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