Answer :
To multiply the polynomials [tex]\((4x^2 + 3x + 7)(8x - 5)\)[/tex], we will use the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms.
1. Distribute the [tex]\(4x^2\)[/tex] term:
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
4x^2 \times 8x = 32x^3
\][/tex]
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
4x^2 \times -5 = -20x^2
\][/tex]
2. Distribute the [tex]\(3x\)[/tex] term:
- Multiply [tex]\(3x\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
3x \times 8x = 24x^2
\][/tex]
- Multiply [tex]\(3x\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
3x \times -5 = -15x
\][/tex]
3. Distribute the [tex]\(7\)[/tex] term:
- Multiply [tex]\(7\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
7 \times 8x = 56x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
7 \times -5 = -35
\][/tex]
4. Combine all the terms:
- Collecting all the terms together, we have:
[tex]\[
32x^3 + (-20x^2) + 24x^2 + (-15x) + 56x + (-35)
\][/tex]
5. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-20x^2 + 24x^2 = 4x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-15x + 56x = 41x
\][/tex]
- The constant term remains [tex]\(-35\)[/tex].
After combining, the final expression is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Therefore, the correct answer is: [tex]\(\boxed{32x^3 + 4x^2 + 41x - 35}\)[/tex]
The answer choice that matches this expression is option C.
1. Distribute the [tex]\(4x^2\)[/tex] term:
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
4x^2 \times 8x = 32x^3
\][/tex]
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
4x^2 \times -5 = -20x^2
\][/tex]
2. Distribute the [tex]\(3x\)[/tex] term:
- Multiply [tex]\(3x\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
3x \times 8x = 24x^2
\][/tex]
- Multiply [tex]\(3x\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
3x \times -5 = -15x
\][/tex]
3. Distribute the [tex]\(7\)[/tex] term:
- Multiply [tex]\(7\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
7 \times 8x = 56x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\[
7 \times -5 = -35
\][/tex]
4. Combine all the terms:
- Collecting all the terms together, we have:
[tex]\[
32x^3 + (-20x^2) + 24x^2 + (-15x) + 56x + (-35)
\][/tex]
5. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-20x^2 + 24x^2 = 4x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-15x + 56x = 41x
\][/tex]
- The constant term remains [tex]\(-35\)[/tex].
After combining, the final expression is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Therefore, the correct answer is: [tex]\(\boxed{32x^3 + 4x^2 + 41x - 35}\)[/tex]
The answer choice that matches this expression is option C.
