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Answer :
Let's multiply the polynomials [tex]\((4x^2 + 4x + 6)(7x + 5)\)[/tex] step by step.
1. Distribute each term of the first polynomial with each term of the second polynomial.
We need to multiply every term in the first polynomial by every term in the second polynomial.
[tex]\[
(4x^2 + 4x + 6)(7x + 5) = 4x^2(7x + 5) + 4x(7x + 5) + 6(7x + 5)
\][/tex]
2. Calculate each part.
- First term: Multiply [tex]\(4x^2\)[/tex] by each term in the second polynomial.
[tex]\[
4x^2 \cdot 7x = 28x^3
\][/tex]
[tex]\[
4x^2 \cdot 5 = 20x^2
\][/tex]
- Second term: Multiply [tex]\(4x\)[/tex] by each term in the second polynomial.
[tex]\[
4x \cdot 7x = 28x^2
\][/tex]
[tex]\[
4x \cdot 5 = 20x
\][/tex]
- Third term: Multiply [tex]\(6\)[/tex] by each term in the second polynomial.
[tex]\[
6 \cdot 7x = 42x
\][/tex]
[tex]\[
6 \cdot 5 = 30
\][/tex]
3. Combine all the terms.
Now, let's add all these products together.
[tex]\[
28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30
\][/tex]
4. Combine like terms.
Combine the [tex]\(x^2\)[/tex] and the [tex]\(x\)[/tex] terms:
- For [tex]\(x^2\)[/tex] terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(20x + 42x = 62x\)[/tex]
5. Write the final result.
[tex]\[
28x^3 + 48x^2 + 62x + 30
\][/tex]
So, the correct answer is D. [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].
1. Distribute each term of the first polynomial with each term of the second polynomial.
We need to multiply every term in the first polynomial by every term in the second polynomial.
[tex]\[
(4x^2 + 4x + 6)(7x + 5) = 4x^2(7x + 5) + 4x(7x + 5) + 6(7x + 5)
\][/tex]
2. Calculate each part.
- First term: Multiply [tex]\(4x^2\)[/tex] by each term in the second polynomial.
[tex]\[
4x^2 \cdot 7x = 28x^3
\][/tex]
[tex]\[
4x^2 \cdot 5 = 20x^2
\][/tex]
- Second term: Multiply [tex]\(4x\)[/tex] by each term in the second polynomial.
[tex]\[
4x \cdot 7x = 28x^2
\][/tex]
[tex]\[
4x \cdot 5 = 20x
\][/tex]
- Third term: Multiply [tex]\(6\)[/tex] by each term in the second polynomial.
[tex]\[
6 \cdot 7x = 42x
\][/tex]
[tex]\[
6 \cdot 5 = 30
\][/tex]
3. Combine all the terms.
Now, let's add all these products together.
[tex]\[
28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30
\][/tex]
4. Combine like terms.
Combine the [tex]\(x^2\)[/tex] and the [tex]\(x\)[/tex] terms:
- For [tex]\(x^2\)[/tex] terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(20x + 42x = 62x\)[/tex]
5. Write the final result.
[tex]\[
28x^3 + 48x^2 + 62x + 30
\][/tex]
So, the correct answer is D. [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].
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