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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, also known as the FOIL method for binomials, but extended for polynomials with more terms.
Here's a detailed step-by-step solution:
1. Distribute Each Term:
- Multiply each term in the first polynomial by each term in the second polynomial.
2. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(4x^2 \times 7x = 28x^3\)[/tex]
- [tex]\(4x^2 \times 5 = 20x^2\)[/tex]
3. Multiply [tex]\(4x\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(4x \times 7x = 28x^2\)[/tex]
- [tex]\(4x \times 5 = 20x\)[/tex]
4. Multiply [tex]\(6\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(6 \times 7x = 42x\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]
5. Combine like terms:
- [tex]\(28x^3\)[/tex] (only one term of cubic degree).
- Combine the quadratic terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex].
- Combine the linear terms: [tex]\(20x + 42x = 62x\)[/tex].
- The constant term is [tex]\(30\)[/tex].
So, the result of [tex]\(\left(4x^2 + 4x + 6\right)(7x + 5)\)[/tex] is [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].
Therefore, the correct answer is:
D. [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex]
Here's a detailed step-by-step solution:
1. Distribute Each Term:
- Multiply each term in the first polynomial by each term in the second polynomial.
2. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(4x^2 \times 7x = 28x^3\)[/tex]
- [tex]\(4x^2 \times 5 = 20x^2\)[/tex]
3. Multiply [tex]\(4x\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(4x \times 7x = 28x^2\)[/tex]
- [tex]\(4x \times 5 = 20x\)[/tex]
4. Multiply [tex]\(6\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(6 \times 7x = 42x\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]
5. Combine like terms:
- [tex]\(28x^3\)[/tex] (only one term of cubic degree).
- Combine the quadratic terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex].
- Combine the linear terms: [tex]\(20x + 42x = 62x\)[/tex].
- The constant term is [tex]\(30\)[/tex].
So, the result of [tex]\(\left(4x^2 + 4x + 6\right)(7x + 5)\)[/tex] is [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].
Therefore, the correct answer is:
D. [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex]
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