Answer :
To find the product of the expressions [tex]\((7x^2)(2x^3+5)(x^2-4x-9)\)[/tex], we can approach this as multiplying these three expressions step by step:
1. Multiply the First Two Expressions:
Start with the expressions [tex]\(7x^2\)[/tex] and [tex]\(2x^3 + 5\)[/tex].
[tex]\[
(7x^2)(2x^3 + 5) = (7x^2 \cdot 2x^3) + (7x^2 \cdot 5)
\][/tex]
- The first part is: [tex]\(7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5\)[/tex].
- The second part is: [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex].
So, the result of this multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply the Result by the Third Expression:
Now, take the result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it by the third expression [tex]\(x^2 - 4x - 9\)[/tex].
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
Distribute each term in [tex]\(14x^5 + 35x^2\)[/tex] across each term in [tex]\(x^2 - 4x - 9\)[/tex]:
- [tex]\(14x^5 \cdot x^2 = 14x^{5+2} = 14x^7\)[/tex]
- [tex]\(14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6\)[/tex]
- [tex]\(14x^5 \cdot (-9) = -126x^5\)[/tex]
- [tex]\(35x^2 \cdot x^2 = 35x^{2+2} = 35x^4\)[/tex]
- [tex]\(35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3\)[/tex]
- [tex]\(35x^2 \cdot (-9) = -315x^2\)[/tex]
Combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This expression matches the third option given in the original question. Therefore, the product of the given expressions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
1. Multiply the First Two Expressions:
Start with the expressions [tex]\(7x^2\)[/tex] and [tex]\(2x^3 + 5\)[/tex].
[tex]\[
(7x^2)(2x^3 + 5) = (7x^2 \cdot 2x^3) + (7x^2 \cdot 5)
\][/tex]
- The first part is: [tex]\(7x^2 \cdot 2x^3 = 14x^{2+3} = 14x^5\)[/tex].
- The second part is: [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex].
So, the result of this multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
2. Multiply the Result by the Third Expression:
Now, take the result [tex]\(14x^5 + 35x^2\)[/tex] and multiply it by the third expression [tex]\(x^2 - 4x - 9\)[/tex].
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
Distribute each term in [tex]\(14x^5 + 35x^2\)[/tex] across each term in [tex]\(x^2 - 4x - 9\)[/tex]:
- [tex]\(14x^5 \cdot x^2 = 14x^{5+2} = 14x^7\)[/tex]
- [tex]\(14x^5 \cdot (-4x) = -56x^{5+1} = -56x^6\)[/tex]
- [tex]\(14x^5 \cdot (-9) = -126x^5\)[/tex]
- [tex]\(35x^2 \cdot x^2 = 35x^{2+2} = 35x^4\)[/tex]
- [tex]\(35x^2 \cdot (-4x) = -140x^{2+1} = -140x^3\)[/tex]
- [tex]\(35x^2 \cdot (-9) = -315x^2\)[/tex]
Combine all these terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This expression matches the third option given in the original question. Therefore, the product of the given expressions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
