High School

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 - 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 expressions [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we'll follow these steps:

1. Multiply the First Two Expressions:

Let's start by multiplying [tex]\(7x^2\)[/tex] and [tex]\(2x^3 + 5\)[/tex]:

- Distribute [tex]\(7x^2\)[/tex] across each term in [tex]\(2x^3 + 5\)[/tex]:
[tex]\[
7x^2 \cdot 2x^3 = 14x^5
\][/tex]
[tex]\[
7x^2 \cdot 5 = 35x^2
\][/tex]

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

2. Multiply the Result with the Third Expression:

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

- Distribute [tex]\(14x^5\)[/tex] across each term in [tex]\(x^2 - 4x - 9\)[/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]

- Then, distribute [tex]\(35x^2\)[/tex] across each term in [tex]\(x^2 - 4x - 9\)[/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 Like Terms:

Now, combine all these results:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

This is the expanded product of the original expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

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