Answer :
To find the product of the given expression [tex]\(\left(7 x^2\right)\left(2 x^3+5\right)\left(x^2-4 x-9\right)\)[/tex], we'll multiply the expressions step by step.
### Step 1: Multiply the first two expressions
First, we'll multiply [tex]\((7x^2)\)[/tex] by [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculate each term:
- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, the result of this multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result by the third expression
Now, multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
Distribute each term in the first polynomial to each term in the second polynomial:
1. [tex]\(14x^5 \times x^2 = 14x^7\)[/tex]
2. [tex]\(14x^5 \times (-4x) = -56x^6\)[/tex]
3. [tex]\(14x^5 \times (-9) = -126x^5\)[/tex]
4. [tex]\(35x^2 \times x^2 = 35x^4\)[/tex]
5. [tex]\(35x^2 \times (-4x) = -140x^3\)[/tex]
6. [tex]\(35x^2 \times (-9) = -315x^2\)[/tex]
Combine all these results:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the final product is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
### Step 1: Multiply the first two expressions
First, we'll multiply [tex]\((7x^2)\)[/tex] by [tex]\((2x^3 + 5)\)[/tex]:
[tex]\[
7x^2 \times (2x^3 + 5) = 7x^2 \cdot 2x^3 + 7x^2 \cdot 5
\][/tex]
Calculate each term:
- [tex]\(7x^2 \cdot 2x^3 = 14x^5\)[/tex]
- [tex]\(7x^2 \cdot 5 = 35x^2\)[/tex]
So, the result of this multiplication is:
[tex]\[
14x^5 + 35x^2
\][/tex]
### Step 2: Multiply the result by the third expression
Now, multiply [tex]\((14x^5 + 35x^2)\)[/tex] by [tex]\((x^2 - 4x - 9)\)[/tex]:
[tex]\[
(14x^5 + 35x^2) \times (x^2 - 4x - 9)
\][/tex]
Distribute each term in the first polynomial to each term in the second polynomial:
1. [tex]\(14x^5 \times x^2 = 14x^7\)[/tex]
2. [tex]\(14x^5 \times (-4x) = -56x^6\)[/tex]
3. [tex]\(14x^5 \times (-9) = -126x^5\)[/tex]
4. [tex]\(35x^2 \times x^2 = 35x^4\)[/tex]
5. [tex]\(35x^2 \times (-4x) = -140x^3\)[/tex]
6. [tex]\(35x^2 \times (-9) = -315x^2\)[/tex]
Combine all these results:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
Therefore, the final product is:
[tex]\[
\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}
\][/tex]
