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What is the product of the following expression?

[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 given expressions [tex]\(\left(7x^2\right)\left(2x^3+5\right)\left(x^2-4x-9\right)\)[/tex], let's take it one step at a time.

### Step 1: Multiply the first two expressions

First, we'll multiply:
[tex]\[
7x^2 \cdot (2x^3 + 5)
\][/tex]

Distribute [tex]\(7x^2\)[/tex] across the terms in the parentheses:

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

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

### Step 2: Multiply the result by the third expression

Now, we need to multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].

Let's distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across [tex]\((x^2 - 4x - 9)\)[/tex].

#### Distribute [tex]\(14x^5\)[/tex]:
- [tex]\(14x^5 \times x^2 = 14x^7\)[/tex]
- [tex]\(14x^5 \times (-4x) = -56x^6\)[/tex]
- [tex]\(14x^5 \times (-9) = -126x^5\)[/tex]

#### Distribute [tex]\(35x^2\)[/tex]:
- [tex]\(35x^2 \times x^2 = 35x^4\)[/tex]
- [tex]\(35x^2 \times (-4x) = -140x^3\)[/tex]
- [tex]\(35x^2 \times (-9) = -315x^2\)[/tex]

### Step 3: Combine all terms and simplify

Now, let's write down all these terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]

Combine like terms (if any) to ensure the polynomial is simplified. Here, none of the terms are alike, so the simplified final product is:

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

This is the expanded polynomial form of the product of the original expressions.

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