Answer :
Sure, let's break down the problem step by step.
We are given the expression:
[tex]\[
\left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x - 9\right)
\][/tex]
We need to find the product of these three expressions. We'll do this step-by-step.
### Step 1: Multiply the first two expressions
First, we multiply [tex]\((7x^2)\)[/tex] by [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
When we multiply [tex]\(7x^2\)[/tex] by [tex]\(2x^3\)[/tex], we get:
[tex]\[
7 \cdot 2 \cdot x^{2+3} = 14x^5
\][/tex]
When we multiply [tex]\(7x^2\)[/tex] by [tex]\(5\)[/tex], we get:
[tex]\[
7 \cdot 5 \cdot x^2 = 35x^2
\][/tex]
So, the product of the first two expressions is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the resulting expression by the third expression
Now we multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
We use the distributive property (also known as the FOIL method for these kinds of expressions) to expand this product:
[tex]\[
14x^5(x^2) + 14x^5(-4x) + 14x^5(-9) + 35x^2(x^2) + 35x^2(-4x) + 35x^2(-9)
\][/tex]
Now, we multiply each term:
1. [tex]\(14x^5 \cdot x^2 = 14x^{5+2} = 14x^7\)[/tex]
2. [tex]\(14x^5 \cdot (-4x) = -14 \cdot 4 \cdot x^{5+1} = -56x^6\)[/tex]
3. [tex]\(14x^5 \cdot (-9) = -14 \cdot 9 \cdot x^5 = -126x^5\)[/tex]
4. [tex]\(35x^2 \cdot x^2 = 35 \cdot x^{2+2} = 35x^4\)[/tex]
5. [tex]\(35x^2 \cdot (-4x) = -35 \cdot 4 \cdot x^{2+1} = -140x^3\)[/tex]
6. [tex]\(35x^2 \cdot (-9) = -35 \cdot 9 \cdot x^2 = -315x^2\)[/tex]
Finally, we sum all these terms to get the final expanded expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the product of the given expressions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
### Conclusion
After carefully expanding and combining like terms, we find that the correct product of the given expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
The third option in the problem corresponds to this result.
We are given the expression:
[tex]\[
\left(7x^2\right)\left(2x^3 + 5\right)\left(x^2 - 4x - 9\right)
\][/tex]
We need to find the product of these three expressions. We'll do this step-by-step.
### Step 1: Multiply the first two expressions
First, we multiply [tex]\((7x^2)\)[/tex] by [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
(7x^2)(2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
When we multiply [tex]\(7x^2\)[/tex] by [tex]\(2x^3\)[/tex], we get:
[tex]\[
7 \cdot 2 \cdot x^{2+3} = 14x^5
\][/tex]
When we multiply [tex]\(7x^2\)[/tex] by [tex]\(5\)[/tex], we get:
[tex]\[
7 \cdot 5 \cdot x^2 = 35x^2
\][/tex]
So, the product of the first two expressions is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the resulting expression by the third expression
Now we multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2)(x^2 - 4x - 9)
\][/tex]
We use the distributive property (also known as the FOIL method for these kinds of expressions) to expand this product:
[tex]\[
14x^5(x^2) + 14x^5(-4x) + 14x^5(-9) + 35x^2(x^2) + 35x^2(-4x) + 35x^2(-9)
\][/tex]
Now, we multiply each term:
1. [tex]\(14x^5 \cdot x^2 = 14x^{5+2} = 14x^7\)[/tex]
2. [tex]\(14x^5 \cdot (-4x) = -14 \cdot 4 \cdot x^{5+1} = -56x^6\)[/tex]
3. [tex]\(14x^5 \cdot (-9) = -14 \cdot 9 \cdot x^5 = -126x^5\)[/tex]
4. [tex]\(35x^2 \cdot x^2 = 35 \cdot x^{2+2} = 35x^4\)[/tex]
5. [tex]\(35x^2 \cdot (-4x) = -35 \cdot 4 \cdot x^{2+1} = -140x^3\)[/tex]
6. [tex]\(35x^2 \cdot (-9) = -35 \cdot 9 \cdot x^2 = -315x^2\)[/tex]
Finally, we sum all these terms to get the final expanded expression:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
So, the product of the given expressions is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
### Conclusion
After carefully expanding and combining like terms, we find that the correct product of the given expression is:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
The third option in the problem corresponds to this result.
