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Answer :
To solve the problem of finding the product of the expression [tex]\((7x^2)(2x^3+5)(x^2-4x-9)\)[/tex], we need to expand the expression step by step.
Here's how we can do it:
1. Identify the Product:
The given expression is [tex]\((7x^2)\cdot(2x^3+5)\cdot(x^2-4x-9)\)[/tex].
2. Step-by-Step Expansion:
- First, focus on the first two parts: [tex]\((7x^2)(2x^3 + 5)\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by each term inside the parentheses:
- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
- Result: [tex]\(14x^5 + 35x^2\)[/tex].
- Next, multiply this result by the third part: [tex]\((x^2 - 4x - 9)\)[/tex].
- Multiply [tex]\(14x^5\)[/tex] by 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]
- Multiply [tex]\(35x^2\)[/tex] by 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, add all the terms together:
- [tex]\(14x^7\)[/tex]
- [tex]\(-56x^6\)[/tex]
- [tex]\(-126x^5\)[/tex]
- [tex]\(+35x^4\)[/tex]
- [tex]\(-140x^3\)[/tex]
- [tex]\(-315x^2\)[/tex]
The combined result is:
[tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex].
4. Match with Options:
This matches the third option from the given choices:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the correct product of the expression is the third option.
Here's how we can do it:
1. Identify the Product:
The given expression is [tex]\((7x^2)\cdot(2x^3+5)\cdot(x^2-4x-9)\)[/tex].
2. Step-by-Step Expansion:
- First, focus on the first two parts: [tex]\((7x^2)(2x^3 + 5)\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by each term inside the parentheses:
- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
- Result: [tex]\(14x^5 + 35x^2\)[/tex].
- Next, multiply this result by the third part: [tex]\((x^2 - 4x - 9)\)[/tex].
- Multiply [tex]\(14x^5\)[/tex] by 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]
- Multiply [tex]\(35x^2\)[/tex] by 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, add all the terms together:
- [tex]\(14x^7\)[/tex]
- [tex]\(-56x^6\)[/tex]
- [tex]\(-126x^5\)[/tex]
- [tex]\(+35x^4\)[/tex]
- [tex]\(-140x^3\)[/tex]
- [tex]\(-315x^2\)[/tex]
The combined result is:
[tex]\(14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2\)[/tex].
4. Match with Options:
This matches the third option from the given choices:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the correct product of the expression is the third option.
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