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What is the product of the following expression?

[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]

Answer :

To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we need to follow these multiplication steps:

1. Multiply [tex]\(7x^2\)[/tex] with [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \cdot (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
[tex]\[
= 14x^5 + 35x^2
\][/tex]

2. Now multiply the result [tex]\((14x^5 + 35x^2)\)[/tex] with [tex]\((x^2 - 4x - 9)\)[/tex]:
- 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]

- 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]

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

So, the final product is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This is the complete expanded form of the given expression.

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