Answer :
To find the product of the expression [tex]\((7x^2)(2x^3 + 5)(x^2 - 4x - 9)\)[/tex], we can expand it step-by-step.
### Step 1: Expand [tex]\( (2x^3 + 5)(x^2 - 4x - 9) \)[/tex]
Distribute each term in the first polynomial to each term in the second polynomial:
1. Multiply [tex]\(2x^3\)[/tex] by each term in [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]
2. Multiply [tex]\(5\)[/tex] by each term in [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]
Combine all these terms together:
[tex]\[ 2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45 \][/tex]
### Step 2: Multiply the result by [tex]\(7x^2\)[/tex]
Now, multiply each term in [tex]\(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45\)[/tex] by [tex]\(7x^2\)[/tex]:
1. [tex]\(7x^2 \cdot 2x^5 = 14x^7\)[/tex]
2. [tex]\(7x^2 \cdot (-8x^4) = -56x^6\)[/tex]
3. [tex]\(7x^2 \cdot (-18x^3) = -126x^5\)[/tex]
4. [tex]\(7x^2 \cdot 5x^2 = 35x^4\)[/tex]
5. [tex]\(7x^2 \cdot (-20x) = -140x^3\)[/tex]
6. [tex]\(7x^2 \cdot (-45) = -315x^2\)[/tex]
Combine all these terms for the final result:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
So, the product of the expression is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
### Step 1: Expand [tex]\( (2x^3 + 5)(x^2 - 4x - 9) \)[/tex]
Distribute each term in the first polynomial to each term in the second polynomial:
1. Multiply [tex]\(2x^3\)[/tex] by each term in [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]
2. Multiply [tex]\(5\)[/tex] by each term in [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]
Combine all these terms together:
[tex]\[ 2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45 \][/tex]
### Step 2: Multiply the result by [tex]\(7x^2\)[/tex]
Now, multiply each term in [tex]\(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45\)[/tex] by [tex]\(7x^2\)[/tex]:
1. [tex]\(7x^2 \cdot 2x^5 = 14x^7\)[/tex]
2. [tex]\(7x^2 \cdot (-8x^4) = -56x^6\)[/tex]
3. [tex]\(7x^2 \cdot (-18x^3) = -126x^5\)[/tex]
4. [tex]\(7x^2 \cdot 5x^2 = 35x^4\)[/tex]
5. [tex]\(7x^2 \cdot (-20x) = -140x^3\)[/tex]
6. [tex]\(7x^2 \cdot (-45) = -315x^2\)[/tex]
Combine all these terms for the final result:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
So, the product of the expression is:
[tex]\[ 14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2 \][/tex]
