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

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

A. $ 14x^5 - x^4 - 46x^3 - 58x^2 - 20x - 45 $

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

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

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

We want to simplify the product

[tex]$$
\left(7x^2\right)\left(2x^3+5\right)\left(x^2-4x-9\right).
$$[/tex]

We'll break the process into two main steps.

---

Step 1. Multiply the first two factors

Multiply the two factors:

[tex]$$
7x^2 \quad \text{and} \quad 2x^3 + 5.
$$[/tex]

Distribute [tex]$7x^2$[/tex] over the sum:

[tex]\[
7x^2 \cdot (2x^3) = 14x^5,
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2.
\][/tex]

So their product is:

[tex]$$
14x^5 + 35x^2.
$$[/tex]

---

Step 2. Multiply the result by the third factor

Now multiply the result

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

by

[tex]$$
x^2 - 4x - 9.
$$[/tex]

Distribute each term in [tex]$14x^5 + 35x^2$[/tex] to every term in [tex]$x^2 - 4x - 9$[/tex].

* Multiply [tex]$14x^5$[/tex] by each term:

[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] by each term:

[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]

Now, combine all the resulting terms:

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

This is the product in expanded form.

---

Final Answer:

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