Answer :
To solve the expression [tex]\((5x^2 + 2x + 8)(7x - 6)\)[/tex], we will use the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial.
Let's do the multiplication step-by-step:
1. First, multiply the term [tex]\(5x^2\)[/tex] with each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times (-6) = -30x^2\)[/tex]
2. Next, multiply the term [tex]\(2x\)[/tex] with each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times (-6) = -12x\)[/tex]
3. Then, multiply the constant [tex]\(8\)[/tex] with each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times (-6) = -48\)[/tex]
Now, combine all the results from the above multiplications:
[tex]\[
35x^3 + (-30x^2) + 14x^2 + (-12x) + 56x + (-48)
\][/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(35x^3\)[/tex].
- 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].
- The constant term is [tex]\(-48\)[/tex].
Putting it all together, the expanded polynomial is:
[tex]\[ 35x^3 - 16x^2 + 44x - 48 \][/tex]
This matches option A: [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
Let's do the multiplication step-by-step:
1. First, multiply the term [tex]\(5x^2\)[/tex] with each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times (-6) = -30x^2\)[/tex]
2. Next, multiply the term [tex]\(2x\)[/tex] with each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times (-6) = -12x\)[/tex]
3. Then, multiply the constant [tex]\(8\)[/tex] with each term in [tex]\((7x - 6)\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times (-6) = -48\)[/tex]
Now, combine all the results from the above multiplications:
[tex]\[
35x^3 + (-30x^2) + 14x^2 + (-12x) + 56x + (-48)
\][/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(35x^3\)[/tex].
- 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].
- The constant term is [tex]\(-48\)[/tex].
Putting it all together, the expanded polynomial is:
[tex]\[ 35x^3 - 16x^2 + 44x - 48 \][/tex]
This matches option A: [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
