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What is the product?

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

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. \[ 14x^{12} - 182x^6 + 35x^4 - 455x^2 \]

Answer :

To find the product of [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we can expand it step-by-step.

### Step 1: Expand [tex]\((2x^3 + 5)(x^2 - 4x - 9)\)[/tex]

First, apply the distributive property to multiply the expressions:

[tex]\[
(2x^3 + 5)(x^2 - 4x - 9) = 2x^3(x^2 - 4x - 9) + 5(x^2 - 4x - 9)
\][/tex]

Expand each part separately:

- [tex]\(2x^3 \cdot x^2 = 2x^5\)[/tex]
- [tex]\(2x^3 \cdot (-4x) = -8x^4\)[/tex]
- [tex]\(2x^3 \cdot (-9) = -18x^3\)[/tex]
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-4x) = -20x\)[/tex]
- [tex]\(5 \cdot (-9) = -45\)[/tex]

Combining these, we get:

[tex]\[
2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45
\][/tex]

### Step 2: Multiply by [tex]\(7x^2\)[/tex]

Now multiply the resulting polynomial by [tex]\(7x^2\)[/tex]:

Apply the distributive property again:

[tex]\[
7x^2(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45)
\][/tex]

Expand each term:

- [tex]\(7x^2 \cdot 2x^5 = 14x^7\)[/tex]
- [tex]\(7x^2 \cdot (-8x^4) = -56x^6\)[/tex]
- [tex]\(7x^2 \cdot (-18x^3) = -126x^5\)[/tex]
- [tex]\(7x^2 \cdot 5x^2 = 35x^4\)[/tex]
- [tex]\(7x^2 \cdot (-20x) = -140x^3\)[/tex]
- [tex]\(7x^2 \cdot (-45) = -315x^2\)[/tex]

Putting all these together gives:

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

This simplifies to:

Answer: [tex]\(\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}\)[/tex]

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Rewritten by : Barada