College

What is the product?

[tex]\[

(7x^2)(2x^3+5)(x^2-4x-9)

\][/tex]

A. [tex]\(14x^5 - x^4 - 46x^3 - 58x^2 - 20x - 46\)[/tex]

B. [tex]\(14x^5 - 56x^5 - 81x^4 - 140x^3 - 315x^2\)[/tex]

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

D. [tex]\(14x^{12} - 182x^5 + 35x^4 - 455x^2\)[/tex]

Answer :

To find the product of the given expressions, follow these steps:

1. Identify Each Expression:

The expressions given are:
- [tex]\( 7x^2 \)[/tex]
- [tex]\( 2x^3 + 5 \)[/tex]
- [tex]\( x^2 - 4x - 9 \)[/tex]

2. Multiply the Expressions Step-by-Step:

First, we multiply the first two expressions:

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

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

So, the result of the first multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]

3. Multiply the Result with the Third Expression:

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

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

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

- [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]

4. Combine Like Terms:

Now add all these results together:

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

This is the expanded form of the product. Thus, the product of the given expressions is:

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

This matches the result provided, confirming the solution.