Answer :
Sure! Let's find the product of the given expression step-by-step.
We want to multiply the following:
[tex]\[ (7x^2)(2x^3 + 5)(x^2 - 4x - 9) \][/tex]
### Step-by-Step Solution:
1. Expand the innermost expressions first:
Start by expanding [tex]\((2x^3 + 5)(x^2 - 4x - 9)\)[/tex].
2. Distribute each term in [tex]\((2x^3 + 5)\)[/tex] to each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
2x^3 (x^2 - 4x - 9) + 5 (x^2 - 4x - 9)
\][/tex]
3. Distribute [tex]\(2x^3\)[/tex] to [tex]\((x^2 - 4x - 9)\)[/tex]:
[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]
So, [tex]\(2x^3 (x^2 - 4x - 9)\)[/tex] becomes:
[tex]\[
2x^5 - 8x^4 - 18x^3
\][/tex]
4. Distribute 5 to [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
5 \cdot x^2 = 5x^2
\][/tex]
[tex]\[
5 \cdot (-4x) = -20x
\][/tex]
[tex]\[
5 \cdot (-9) = -45
\][/tex]
So, [tex]\(5 (x^2 - 4x - 9)\)[/tex] becomes:
[tex]\[
5x^2 - 20x - 45
\][/tex]
5. Combine the results from the two distributions:
[tex]\[
(2x^5 - 8x^4 - 18x^3) + (5x^2 - 20x - 45)
\][/tex]
Combining like terms gives us:
[tex]\[
2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45
\][/tex]
6. Now multiply this result by [tex]\(7x^2\)[/tex]:
[tex]\[
7x^2 \cdot (2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45)
\][/tex]
7. Distribute [tex]\(7x^2\)[/tex] to 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]
8. Combine all the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the product of [tex]\(\left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x - 9\right)\)[/tex] is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
The correct answer is therefore:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
We want to multiply the following:
[tex]\[ (7x^2)(2x^3 + 5)(x^2 - 4x - 9) \][/tex]
### Step-by-Step Solution:
1. Expand the innermost expressions first:
Start by expanding [tex]\((2x^3 + 5)(x^2 - 4x - 9)\)[/tex].
2. Distribute each term in [tex]\((2x^3 + 5)\)[/tex] to each term in [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
2x^3 (x^2 - 4x - 9) + 5 (x^2 - 4x - 9)
\][/tex]
3. Distribute [tex]\(2x^3\)[/tex] to [tex]\((x^2 - 4x - 9)\)[/tex]:
[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]
So, [tex]\(2x^3 (x^2 - 4x - 9)\)[/tex] becomes:
[tex]\[
2x^5 - 8x^4 - 18x^3
\][/tex]
4. Distribute 5 to [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
5 \cdot x^2 = 5x^2
\][/tex]
[tex]\[
5 \cdot (-4x) = -20x
\][/tex]
[tex]\[
5 \cdot (-9) = -45
\][/tex]
So, [tex]\(5 (x^2 - 4x - 9)\)[/tex] becomes:
[tex]\[
5x^2 - 20x - 45
\][/tex]
5. Combine the results from the two distributions:
[tex]\[
(2x^5 - 8x^4 - 18x^3) + (5x^2 - 20x - 45)
\][/tex]
Combining like terms gives us:
[tex]\[
2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45
\][/tex]
6. Now multiply this result by [tex]\(7x^2\)[/tex]:
[tex]\[
7x^2 \cdot (2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45)
\][/tex]
7. Distribute [tex]\(7x^2\)[/tex] to 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]
8. Combine all the terms:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the product of [tex]\(\left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x - 9\right)\)[/tex] is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
The correct answer is therefore:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
