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Answer :
To multiply the polynomials
[tex]$$ (4x^2+4x+6)(7x+5), $$[/tex]
we will use the distributive property (also known as the FOIL method for binomials) by multiplying each term in the first polynomial by each term in the second polynomial, and then combine like terms.
1. Multiply each term of [tex]\(4x^2 + 4x + 6\)[/tex] by [tex]\(7x\)[/tex] and [tex]\(5\)[/tex] separately:
- Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(7x+5\)[/tex]:
[tex]\[
4x^2 \cdot 7x = 28x^3, \quad 4x^2 \cdot 5 = 20x^2.
\][/tex]
- Multiply [tex]\(4x\)[/tex] by each term in [tex]\(7x+5\)[/tex]:
[tex]\[
4x \cdot 7x = 28x^2, \quad 4x \cdot 5 = 20x.
\][/tex]
- Multiply [tex]\(6\)[/tex] by each term in [tex]\(7x+5\)[/tex]:
[tex]\[
6 \cdot 7x = 42x, \quad 6 \cdot 5 = 30.
\][/tex]
2. Now, list all the resulting terms:
[tex]\[
28x^3, \quad 20x^2, \quad 28x^2, \quad 20x, \quad 42x, \quad 30.
\][/tex]
3. Combine like terms:
- The [tex]\(x^3\)[/tex] term:
[tex]\[
28x^3.
\][/tex]
- The [tex]\(x^2\)[/tex] terms:
[tex]\[
20x^2 + 28x^2 = 48x^2.
\][/tex]
- The [tex]\(x\)[/tex] terms:
[tex]\[
20x + 42x = 62x.
\][/tex]
- The constant term:
[tex]\[
30.
\][/tex]
4. Therefore, the product of the two polynomials is:
[tex]$$
28x^3+48x^2+62x+30.
$$[/tex]
Thus, the correct answer is option C.
[tex]$$ (4x^2+4x+6)(7x+5), $$[/tex]
we will use the distributive property (also known as the FOIL method for binomials) by multiplying each term in the first polynomial by each term in the second polynomial, and then combine like terms.
1. Multiply each term of [tex]\(4x^2 + 4x + 6\)[/tex] by [tex]\(7x\)[/tex] and [tex]\(5\)[/tex] separately:
- Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(7x+5\)[/tex]:
[tex]\[
4x^2 \cdot 7x = 28x^3, \quad 4x^2 \cdot 5 = 20x^2.
\][/tex]
- Multiply [tex]\(4x\)[/tex] by each term in [tex]\(7x+5\)[/tex]:
[tex]\[
4x \cdot 7x = 28x^2, \quad 4x \cdot 5 = 20x.
\][/tex]
- Multiply [tex]\(6\)[/tex] by each term in [tex]\(7x+5\)[/tex]:
[tex]\[
6 \cdot 7x = 42x, \quad 6 \cdot 5 = 30.
\][/tex]
2. Now, list all the resulting terms:
[tex]\[
28x^3, \quad 20x^2, \quad 28x^2, \quad 20x, \quad 42x, \quad 30.
\][/tex]
3. Combine like terms:
- The [tex]\(x^3\)[/tex] term:
[tex]\[
28x^3.
\][/tex]
- The [tex]\(x^2\)[/tex] terms:
[tex]\[
20x^2 + 28x^2 = 48x^2.
\][/tex]
- The [tex]\(x\)[/tex] terms:
[tex]\[
20x + 42x = 62x.
\][/tex]
- The constant term:
[tex]\[
30.
\][/tex]
4. Therefore, the product of the two polynomials is:
[tex]$$
28x^3+48x^2+62x+30.
$$[/tex]
Thus, the correct answer is option C.
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