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Answer :
To multiply the polynomials [tex]\((4x^2 + 4x + 6)\)[/tex] and [tex]\((7x + 5)\)[/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. Let's go through this step by step:
1. Start by multiplying each term in [tex]\((4x^2 + 4x + 6)\)[/tex] by each term in [tex]\((7x + 5)\)[/tex].
- First, multiply [tex]\(4x^2\)[/tex] by both terms in the second polynomial:
- [tex]\(4x^2 \times 7x = 28x^3\)[/tex]
- [tex]\(4x^2 \times 5 = 20x^2\)[/tex]
- Next, multiply [tex]\(4x\)[/tex] by both terms in the second polynomial:
- [tex]\(4x \times 7x = 28x^2\)[/tex]
- [tex]\(4x \times 5 = 20x\)[/tex]
- Finally, multiply [tex]\(6\)[/tex] by both terms in the second polynomial:
- [tex]\(6 \times 7x = 42x\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]
2. Add all the products together:
[tex]\[
28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30
\][/tex]
3. Combine the like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(28x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(20x + 42x = 62x\)[/tex].
- The constant term is [tex]\(30\)[/tex].
So, the resulting polynomial is:
[tex]\[ 28x^3 + 48x^2 + 62x + 30 \][/tex]
The correct answer is option A: [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].
1. Start by multiplying each term in [tex]\((4x^2 + 4x + 6)\)[/tex] by each term in [tex]\((7x + 5)\)[/tex].
- First, multiply [tex]\(4x^2\)[/tex] by both terms in the second polynomial:
- [tex]\(4x^2 \times 7x = 28x^3\)[/tex]
- [tex]\(4x^2 \times 5 = 20x^2\)[/tex]
- Next, multiply [tex]\(4x\)[/tex] by both terms in the second polynomial:
- [tex]\(4x \times 7x = 28x^2\)[/tex]
- [tex]\(4x \times 5 = 20x\)[/tex]
- Finally, multiply [tex]\(6\)[/tex] by both terms in the second polynomial:
- [tex]\(6 \times 7x = 42x\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]
2. Add all the products together:
[tex]\[
28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30
\][/tex]
3. Combine the like terms:
- The [tex]\(x^3\)[/tex] term is [tex]\(28x^3\)[/tex].
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(20x + 42x = 62x\)[/tex].
- The constant term is [tex]\(30\)[/tex].
So, the resulting polynomial is:
[tex]\[ 28x^3 + 48x^2 + 62x + 30 \][/tex]
The correct answer is option A: [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].
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