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 expression [tex]$(7x^2)(2x^3 + 5)(x^2 - 4x - 9)$[/tex], we need to multiply the three polynomials together. Here is a step-by-step breakdown of how we can do this:

1. Multiply the first two expressions:
[tex]\[
(7x^2)(2x^3 + 5)
\][/tex]
Distribute [tex]$7x^2$[/tex] over each term in the second polynomial:
[tex]\[
= 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Simplify each term:
[tex]\[
= 14x^5 + 35x^2
\][/tex]

2. Multiply the result by the third expression:
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]

3. Distribute [tex]$14x^5$[/tex] over the third 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]

4. Distribute [tex]$35x^2$[/tex] over the third polynomial:
[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]

5. Combine all the terms:
[tex]\[
= 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

By carefully multiplying each of the terms, we arrive at the final expanded product of the polynomial. The complete solution in the expanded form is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]