Answer :
Sure! Let's multiply the polynomials [tex]\((8x^2 + 6x + 8)\)[/tex] and [tex]\((6x - 5)\)[/tex] step-by-step.
### Step 1: Distribute Each Term
1. Multiply [tex]\(8x^2\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(8x^2 \cdot 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \cdot (-5) = -40x^2\)[/tex]
2. Multiply [tex]\(6x\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot (-5) = -30x\)[/tex]
3. Multiply [tex]\(8\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot (-5) = -40\)[/tex]
### Step 2: Combine Like Terms
- We'll add up all the terms you got from the distributions:
1. [tex]\(x^3\)[/tex] term:
- [tex]\(48x^3\)[/tex]
2. [tex]\(x^2\)[/tex] terms:
- Combine [tex]\(-40x^2\)[/tex] and [tex]\(36x^2\)[/tex] to get [tex]\(-4x^2\)[/tex]
3. [tex]\(x\)[/tex] terms:
- Combine [tex]\(-30x\)[/tex] and [tex]\(48x\)[/tex] to get [tex]\(18x\)[/tex]
4. Constant term:
- [tex]\(-40\)[/tex]
### Final Polynomial
Putting it all together, the product of the polynomials is:
[tex]\[ 48x^3 - 4x^2 + 18x - 40 \][/tex]
So, the correct option is D. [tex]\( 48x^3 - 4x^2 + 18x - 40 \)[/tex].
### Step 1: Distribute Each Term
1. Multiply [tex]\(8x^2\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(8x^2 \cdot 6x = 48x^3\)[/tex]
- [tex]\(8x^2 \cdot (-5) = -40x^2\)[/tex]
2. Multiply [tex]\(6x\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(6x \cdot 6x = 36x^2\)[/tex]
- [tex]\(6x \cdot (-5) = -30x\)[/tex]
3. Multiply [tex]\(8\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- [tex]\(8 \cdot 6x = 48x\)[/tex]
- [tex]\(8 \cdot (-5) = -40\)[/tex]
### Step 2: Combine Like Terms
- We'll add up all the terms you got from the distributions:
1. [tex]\(x^3\)[/tex] term:
- [tex]\(48x^3\)[/tex]
2. [tex]\(x^2\)[/tex] terms:
- Combine [tex]\(-40x^2\)[/tex] and [tex]\(36x^2\)[/tex] to get [tex]\(-4x^2\)[/tex]
3. [tex]\(x\)[/tex] terms:
- Combine [tex]\(-30x\)[/tex] and [tex]\(48x\)[/tex] to get [tex]\(18x\)[/tex]
4. Constant term:
- [tex]\(-40\)[/tex]
### Final Polynomial
Putting it all together, the product of the polynomials is:
[tex]\[ 48x^3 - 4x^2 + 18x - 40 \][/tex]
So, the correct option is D. [tex]\( 48x^3 - 4x^2 + 18x - 40 \)[/tex].
