Answer :
To find the product of the given expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we need to multiply these three expressions together. I'll walk you through the steps to arrive at the final product.
### Step 1: Multiply the first two expressions
First, we'll multiply [tex]\(7x^2\)[/tex] by [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]
Adding these results together gives:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result with the third expression
Now, we need to multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across every term in [tex]\((x^2 - 4x - 9)\)[/tex]:
1. 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]
2. 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]
### Step 3: Combine all the terms
Now, combine all like terms from these results:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded form and the product of the original expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex].
### Step 1: Multiply the first two expressions
First, we'll multiply [tex]\(7x^2\)[/tex] by [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]
Adding these results together gives:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result with the third expression
Now, we need to multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex].
Distribute each term in [tex]\((14x^5 + 35x^2)\)[/tex] across every term in [tex]\((x^2 - 4x - 9)\)[/tex]:
1. 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]
2. 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]
### Step 3: Combine all the terms
Now, combine all like terms from these results:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This is the expanded form and the product of the original expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex].
