Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], let's approach it step-by-step.
### Step 1: Expand the Expressions
First, we'll expand [tex]\((2x^3 + 5)(x^2 - 4x - 9)\)[/tex].
1. Distribute [tex]\(2x^3\)[/tex] across the second expression:
- [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]
2. Then distribute [tex]\(5\)[/tex]:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-4x) = -20x\)[/tex]
- [tex]\(5 \cdot (-9) = -45\)[/tex]
Combine all terms:
[tex]\[2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45\][/tex]
### Step 2: Multiply by [tex]\(7x^2\)[/tex]
Now, multiply this result by [tex]\(7x^2\)[/tex]:
1. [tex]\(7x^2 \cdot 2x^5 = 14x^7\)[/tex]
2. [tex]\(7x^2 \cdot (-8x^4) = -56x^6\)[/tex]
3. [tex]\(7x^2 \cdot (-18x^3) = -126x^5\)[/tex]
4. [tex]\(7x^2 \cdot 5x^2 = 35x^4\)[/tex]
5. [tex]\(7x^2 \cdot (-20x) = -140x^3\)[/tex]
6. [tex]\(7x^2 \cdot (-45) = -315x^2\)[/tex]
Combine all terms to get the final expanded expression:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
### Conclusion
The expanded polynomial matches one of the given choices, specifically:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
Thus, the correct choice for the product of the expression is:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
This matches choice 3 in the list of answers.
### Step 1: Expand the Expressions
First, we'll expand [tex]\((2x^3 + 5)(x^2 - 4x - 9)\)[/tex].
1. Distribute [tex]\(2x^3\)[/tex] across the second expression:
- [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]
2. Then distribute [tex]\(5\)[/tex]:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-4x) = -20x\)[/tex]
- [tex]\(5 \cdot (-9) = -45\)[/tex]
Combine all terms:
[tex]\[2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45\][/tex]
### Step 2: Multiply by [tex]\(7x^2\)[/tex]
Now, multiply this result by [tex]\(7x^2\)[/tex]:
1. [tex]\(7x^2 \cdot 2x^5 = 14x^7\)[/tex]
2. [tex]\(7x^2 \cdot (-8x^4) = -56x^6\)[/tex]
3. [tex]\(7x^2 \cdot (-18x^3) = -126x^5\)[/tex]
4. [tex]\(7x^2 \cdot 5x^2 = 35x^4\)[/tex]
5. [tex]\(7x^2 \cdot (-20x) = -140x^3\)[/tex]
6. [tex]\(7x^2 \cdot (-45) = -315x^2\)[/tex]
Combine all terms to get the final expanded expression:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
### Conclusion
The expanded polynomial matches one of the given choices, specifically:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
Thus, the correct choice for the product of the expression is:
[tex]\[14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\][/tex]
This matches choice 3 in the list of answers.
