Answer :
To find the product of [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]
First, apply the distributive property to multiply the expressions:
[tex]\[
(2x^3 + 5)(x^2 - 4x - 9) = 2x^3(x^2 - 4x - 9) + 5(x^2 - 4x - 9)
\][/tex]
Expand each part separately:
- [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]
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-4x) = -20x\)[/tex]
- [tex]\(5 \cdot (-9) = -45\)[/tex]
Combining these, we get:
[tex]\[
2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45
\][/tex]
### Step 2: Multiply by [tex]\(7x^2\)[/tex]
Now multiply the resulting polynomial by [tex]\(7x^2\)[/tex]:
Apply the distributive property again:
[tex]\[
7x^2(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45)
\][/tex]
Expand each term:
- [tex]\(7x^2 \cdot 2x^5 = 14x^7\)[/tex]
- [tex]\(7x^2 \cdot (-8x^4) = -56x^6\)[/tex]
- [tex]\(7x^2 \cdot (-18x^3) = -126x^5\)[/tex]
- [tex]\(7x^2 \cdot 5x^2 = 35x^4\)[/tex]
- [tex]\(7x^2 \cdot (-20x) = -140x^3\)[/tex]
- [tex]\(7x^2 \cdot (-45) = -315x^2\)[/tex]
Putting all these together gives:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This simplifies to:
Answer: [tex]\(\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}\)[/tex]
### Step 1: Expand [tex]\((2x^3 + 5)(x^2 - 4x - 9)\)[/tex]
First, apply the distributive property to multiply the expressions:
[tex]\[
(2x^3 + 5)(x^2 - 4x - 9) = 2x^3(x^2 - 4x - 9) + 5(x^2 - 4x - 9)
\][/tex]
Expand each part separately:
- [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]
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-4x) = -20x\)[/tex]
- [tex]\(5 \cdot (-9) = -45\)[/tex]
Combining these, we get:
[tex]\[
2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45
\][/tex]
### Step 2: Multiply by [tex]\(7x^2\)[/tex]
Now multiply the resulting polynomial by [tex]\(7x^2\)[/tex]:
Apply the distributive property again:
[tex]\[
7x^2(2x^5 - 8x^4 - 18x^3 + 5x^2 - 20x - 45)
\][/tex]
Expand each term:
- [tex]\(7x^2 \cdot 2x^5 = 14x^7\)[/tex]
- [tex]\(7x^2 \cdot (-8x^4) = -56x^6\)[/tex]
- [tex]\(7x^2 \cdot (-18x^3) = -126x^5\)[/tex]
- [tex]\(7x^2 \cdot 5x^2 = 35x^4\)[/tex]
- [tex]\(7x^2 \cdot (-20x) = -140x^3\)[/tex]
- [tex]\(7x^2 \cdot (-45) = -315x^2\)[/tex]
Putting all these together gives:
[tex]\[
14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2
\][/tex]
This simplifies to:
Answer: [tex]\(\boxed{14x^7 - 56x^6 - 126x^5 + 35x^4 - 140x^3 - 315x^2}\)[/tex]
