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Answer :
To find the product of the expressions [tex]\((7x^2)\)[/tex], [tex]\((2x^3 + 5)\)[/tex], and [tex]\((x^2 - 4x - 9)\)[/tex], we'll multiply these polynomials together step by step.
### Step-by-Step Solution:
1. Start with the first and second expressions:
[tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] to 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]
- The result of this multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Now, multiply the result from step 1 with the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
- Multiply each term in [tex]\((14x^5 + 35x^2)\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex], and then add the results together.
- Distribute [tex]\(14x^5\)[/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]
- Distribute [tex]\(35x^2\)[/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 all terms:
- Write all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the product of the given expressions:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
### Step-by-Step Solution:
1. Start with the first and second expressions:
[tex]\((7x^2)\)[/tex] and [tex]\((2x^3 + 5)\)[/tex].
- Distribute [tex]\(7x^2\)[/tex] to 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]
- The result of this multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Now, multiply the result from step 1 with the third expression [tex]\((x^2 - 4x - 9)\)[/tex]:
- Multiply each term in [tex]\((14x^5 + 35x^2)\)[/tex] by each term in [tex]\((x^2 - 4x - 9)\)[/tex], and then add the results together.
- Distribute [tex]\(14x^5\)[/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]
- Distribute [tex]\(35x^2\)[/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 all terms:
- Write all the terms together:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the product of the given expressions:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
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