Answer :
To find the product of [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex], we will take it step by step. Follow along carefully as we expand and simplify the expression.
### Step 1: Distribute [tex]\(7 x^2\)[/tex] to [tex]\((2 x^3 + 5)\)[/tex]
First, let's start by distributing [tex]\(7 x^2\)[/tex] to the binomial [tex]\((2 x^3 + 5)\)[/tex].
[tex]\[
7 x^2 \cdot (2 x^3 + 5) = (7 x^2 \cdot 2 x^3) + (7 x^2 \cdot 5)
\][/tex]
Calculate each term:
[tex]\[
7 x^2 \cdot 2 x^3 = 14 x^{2+3} = 14 x^5
\][/tex]
[tex]\[
7 x^2 \cdot 5 = 35 x^2
\][/tex]
So,
[tex]\[
7 x^2 \cdot (2 x^3 + 5) = 14 x^5 + 35 x^2
\][/tex]
### Step 2: Multiply the result by [tex]\((x^2 - 4x - 9)\)[/tex]
Now, we need to multiply [tex]\((14 x^5 + 35 x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
[tex]\[
(14 x^5 + 35 x^2) \cdot (x^2 - 4x - 9)
\][/tex]
We will distribute each term in [tex]\((14 x^5 + 35 x^2)\)[/tex] to each term in [tex]\((x^2 - 4x - 9)\)[/tex].
#### Distribute [tex]\(14 x^5\)[/tex]:
[tex]\[
14 x^5 \cdot x^2 = 14 x^{5+2} = 14 x^7
\][/tex]
[tex]\[
14 x^5 \cdot (-4 x) = -56 x^{5+1} = -56 x^6
\][/tex]
[tex]\[
14 x^5 \cdot (-9) = -126 x^5
\][/tex]
#### Distribute [tex]\(35 x^2\)[/tex]:
[tex]\[
35 x^2 \cdot x^2 = 35 x^{2+2} = 35 x^4
\][/tex]
[tex]\[
35 x^2 \cdot (-4 x) = -140 x^{2+1} = -140 x^3
\][/tex]
[tex]\[
35 x^2 \cdot (-9) = -315 x^2
\][/tex]
### Step 3: Add all the terms to get the final expression
Now, let's combine all the terms we obtained:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
So, the correct product of the expression [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex] is:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
This corresponds to the answer choice:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
### Step 1: Distribute [tex]\(7 x^2\)[/tex] to [tex]\((2 x^3 + 5)\)[/tex]
First, let's start by distributing [tex]\(7 x^2\)[/tex] to the binomial [tex]\((2 x^3 + 5)\)[/tex].
[tex]\[
7 x^2 \cdot (2 x^3 + 5) = (7 x^2 \cdot 2 x^3) + (7 x^2 \cdot 5)
\][/tex]
Calculate each term:
[tex]\[
7 x^2 \cdot 2 x^3 = 14 x^{2+3} = 14 x^5
\][/tex]
[tex]\[
7 x^2 \cdot 5 = 35 x^2
\][/tex]
So,
[tex]\[
7 x^2 \cdot (2 x^3 + 5) = 14 x^5 + 35 x^2
\][/tex]
### Step 2: Multiply the result by [tex]\((x^2 - 4x - 9)\)[/tex]
Now, we need to multiply [tex]\((14 x^5 + 35 x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
[tex]\[
(14 x^5 + 35 x^2) \cdot (x^2 - 4x - 9)
\][/tex]
We will distribute each term in [tex]\((14 x^5 + 35 x^2)\)[/tex] to each term in [tex]\((x^2 - 4x - 9)\)[/tex].
#### Distribute [tex]\(14 x^5\)[/tex]:
[tex]\[
14 x^5 \cdot x^2 = 14 x^{5+2} = 14 x^7
\][/tex]
[tex]\[
14 x^5 \cdot (-4 x) = -56 x^{5+1} = -56 x^6
\][/tex]
[tex]\[
14 x^5 \cdot (-9) = -126 x^5
\][/tex]
#### Distribute [tex]\(35 x^2\)[/tex]:
[tex]\[
35 x^2 \cdot x^2 = 35 x^{2+2} = 35 x^4
\][/tex]
[tex]\[
35 x^2 \cdot (-4 x) = -140 x^{2+1} = -140 x^3
\][/tex]
[tex]\[
35 x^2 \cdot (-9) = -315 x^2
\][/tex]
### Step 3: Add all the terms to get the final expression
Now, let's combine all the terms we obtained:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
So, the correct product of the expression [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex] is:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
This corresponds to the answer choice:
[tex]\[
14 x^7 - 56 x^6 - 126 x^5 + 35 x^4 - 140 x^3 - 315 x^2
\][/tex]
