Answer :
Let's multiply the polynomials [tex]\((5x^2 + 2x + 8)\)[/tex] and [tex]\((7x - 6)\)[/tex] using the distributive property, also known as FOIL (First, Outside, Inside, Last) for binomials, but extended for polynomials.
### Step-by-Step Solution:
1. Distribute the first term of the first polynomial, [tex]\(5x^2\)[/tex], across the second polynomial:
- [tex]\( (5x^2) \times (7x) = 35x^3 \)[/tex]
- [tex]\( (5x^2) \times (-6) = -30x^2 \)[/tex]
2. Distribute the second term of the first polynomial, [tex]\(2x\)[/tex], across the second polynomial:
- [tex]\( (2x) \times (7x) = 14x^2 \)[/tex]
- [tex]\( (2x) \times (-6) = -12x \)[/tex]
3. Distribute the third term of the first polynomial, [tex]\(8\)[/tex], across the second polynomial:
- [tex]\( (8) \times (7x) = 56x \)[/tex]
- [tex]\( (8) \times (-6) = -48 \)[/tex]
4. Combine like terms from all the products:
- [tex]\(35x^3\)[/tex] (no like terms, so it stays as is)
- [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- [tex]\(56x - 12x = 44x\)[/tex]
- [tex]\(-48\)[/tex] (no like terms, so it stays as is)
Therefore, the resulting polynomial after multiplication is:
[tex]\[ 35x^3 - 16x^2 + 44x - 48 \][/tex]
Looking at the given options, the correct answer is:
A. [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
### Step-by-Step Solution:
1. Distribute the first term of the first polynomial, [tex]\(5x^2\)[/tex], across the second polynomial:
- [tex]\( (5x^2) \times (7x) = 35x^3 \)[/tex]
- [tex]\( (5x^2) \times (-6) = -30x^2 \)[/tex]
2. Distribute the second term of the first polynomial, [tex]\(2x\)[/tex], across the second polynomial:
- [tex]\( (2x) \times (7x) = 14x^2 \)[/tex]
- [tex]\( (2x) \times (-6) = -12x \)[/tex]
3. Distribute the third term of the first polynomial, [tex]\(8\)[/tex], across the second polynomial:
- [tex]\( (8) \times (7x) = 56x \)[/tex]
- [tex]\( (8) \times (-6) = -48 \)[/tex]
4. Combine like terms from all the products:
- [tex]\(35x^3\)[/tex] (no like terms, so it stays as is)
- [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- [tex]\(56x - 12x = 44x\)[/tex]
- [tex]\(-48\)[/tex] (no like terms, so it stays as is)
Therefore, the resulting polynomial after multiplication is:
[tex]\[ 35x^3 - 16x^2 + 44x - 48 \][/tex]
Looking at the given options, the correct answer is:
A. [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
