Answer :
Sure! Let's multiply the polynomials [tex]\((5x^2 + 2x + 8)(7x - 6)\)[/tex] using the distributive property, which involves expanding each term in the first polynomial by each term in the second polynomial.
### Step-by-Step Solution:
1. Distribute [tex]\(5x^2\)[/tex]:
- Multiply [tex]\(5x^2\)[/tex] by each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times -6 = -30x^2\)[/tex]
2. Distribute [tex]\(2x\)[/tex]:
- Multiply [tex]\(2x\)[/tex] by each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times -6 = -12x\)[/tex]
3. Distribute [tex]\(8\)[/tex]:
- Multiply [tex]\(8\)[/tex] by each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times -6 = -48\)[/tex]
4. Combine Like Terms:
- Combine the [tex]\(x^2\)[/tex] terms:
- [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms:
- [tex]\(-12x + 56x = 44x\)[/tex]
5. Write the Expanded Polynomial:
- Combine all terms to get the expanded polynomial:
- [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
So, the correct answer is [tex]\(\boxed{B. \; 35x^3 - 16x^2 + 44x - 48}\)[/tex].
### Step-by-Step Solution:
1. Distribute [tex]\(5x^2\)[/tex]:
- Multiply [tex]\(5x^2\)[/tex] by each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times -6 = -30x^2\)[/tex]
2. Distribute [tex]\(2x\)[/tex]:
- Multiply [tex]\(2x\)[/tex] by each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times -6 = -12x\)[/tex]
3. Distribute [tex]\(8\)[/tex]:
- Multiply [tex]\(8\)[/tex] by each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times -6 = -48\)[/tex]
4. Combine Like Terms:
- Combine the [tex]\(x^2\)[/tex] terms:
- [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms:
- [tex]\(-12x + 56x = 44x\)[/tex]
5. Write the Expanded Polynomial:
- Combine all terms to get the expanded polynomial:
- [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
So, the correct answer is [tex]\(\boxed{B. \; 35x^3 - 16x^2 + 44x - 48}\)[/tex].
