Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex], we will use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. Here's the step-by-step solution:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7x^2 \times 4x = 28x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \times (-6) = -42x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
5x \times 4x = 20x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x \times (-6) = -30x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7 \times 4x = 28x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7 \times (-6) = -42
\][/tex]
2. Combine all the products:
[tex]\[
28x^3 - 42x^2 + 20x^2 - 30x + 28x - 42
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 28x = -2x
\][/tex]
4. Write the final polynomial:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{28x^3 - 22x^2 - 2x - 42}
\][/tex]
This corresponds to option A.
1. Multiply each term in the first polynomial by each term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7x^2 \times 4x = 28x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \times (-6) = -42x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
5x \times 4x = 20x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x \times (-6) = -30x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7 \times 4x = 28x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7 \times (-6) = -42
\][/tex]
2. Combine all the products:
[tex]\[
28x^3 - 42x^2 + 20x^2 - 30x + 28x - 42
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 28x = -2x
\][/tex]
4. Write the final polynomial:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{28x^3 - 22x^2 - 2x - 42}
\][/tex]
This corresponds to option A.
