Answer :
To multiply the polynomials [tex]\((4x^2 + 3x + 7)\)[/tex] and [tex]\((8x - 5)\)[/tex], we'll use the distributive property to find each term in the product.
Here's how you do it step by step:
1. Multiply each term in the first polynomial by each term in the second polynomial.
- 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]
- 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]
- 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]
2. Add all the terms together.
Combine like terms:
[tex]\[
32x^3 + (-20x^2 + 24x^2) + (-15x + 56x) - 35
\][/tex]
3. Simplify the expression.
- 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]
So, the result is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Thus, the product of the polynomials is [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex], which corresponds to option D.
Here's how you do it step by step:
1. Multiply each term in the first polynomial by each term in the second polynomial.
- 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]
- 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]
- 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]
2. Add all the terms together.
Combine like terms:
[tex]\[
32x^3 + (-20x^2 + 24x^2) + (-15x + 56x) - 35
\][/tex]
3. Simplify the expression.
- 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]
So, the result is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
Thus, the product of the polynomials is [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex], which corresponds to option D.
