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Answer :
To multiply the polynomials
[tex]$$
(4x^2 + 4x + 6)(7x + 5),
$$[/tex]
we follow these steps:
1. Multiply each term from the first polynomial by each term of the second polynomial:
- Multiply the leading terms:
[tex]$$
4x^2 \cdot 7x = 28x^3.
$$[/tex]
- Multiply [tex]$4x^2$[/tex] by [tex]$5$[/tex]:
[tex]$$
4x^2 \cdot 5 = 20x^2.
$$[/tex]
- Multiply [tex]$4x$[/tex] by [tex]$7x$[/tex]:
[tex]$$$
4x \cdot 7x = 28x^2.
$$[/tex]
- Multiply [tex]$4x$[/tex] by [tex]$5$[/tex]:
[tex]$$
4x \cdot 5 = 20x.
$$[/tex]
- Multiply [tex]$6$[/tex] by [tex]$7x$[/tex]:
[tex]$$
6 \cdot 7x = 42x.
$$[/tex]
- Multiply [tex]$6$[/tex] by [tex]$5$[/tex]:
[tex]$$
6 \cdot 5 = 30.
$$[/tex]
2. Group and combine the like terms (terms with the same power of [tex]$x$[/tex]):
- For the [tex]$x^3$[/tex] term:
[tex]$$
28x^3.
$$[/tex]
- For the [tex]$x^2$[/tex] term, add [tex]$20x^2$[/tex] and [tex]$28x^2$[/tex]:
[tex]$$
20x^2 + 28x^2 = 48x^2.
$$[/tex]
- For the [tex]$x$[/tex] term, add [tex]$20x$[/tex] and [tex]$42x$[/tex]:
[tex]$$
20x + 42x = 62x.
$$[/tex]
- The constant term remains:
[tex]$$
30.
$$[/tex]
3. Write the final resulting polynomial:
[tex]$$
28x^3 + 48x^2 + 62x + 30.
$$[/tex]
Thus, the correct answer is:
[tex]$$
\boxed{28x^3 + 48x^2 + 62x + 30}.
$$[/tex]
[tex]$$
(4x^2 + 4x + 6)(7x + 5),
$$[/tex]
we follow these steps:
1. Multiply each term from the first polynomial by each term of the second polynomial:
- Multiply the leading terms:
[tex]$$
4x^2 \cdot 7x = 28x^3.
$$[/tex]
- Multiply [tex]$4x^2$[/tex] by [tex]$5$[/tex]:
[tex]$$
4x^2 \cdot 5 = 20x^2.
$$[/tex]
- Multiply [tex]$4x$[/tex] by [tex]$7x$[/tex]:
[tex]$$$
4x \cdot 7x = 28x^2.
$$[/tex]
- Multiply [tex]$4x$[/tex] by [tex]$5$[/tex]:
[tex]$$
4x \cdot 5 = 20x.
$$[/tex]
- Multiply [tex]$6$[/tex] by [tex]$7x$[/tex]:
[tex]$$
6 \cdot 7x = 42x.
$$[/tex]
- Multiply [tex]$6$[/tex] by [tex]$5$[/tex]:
[tex]$$
6 \cdot 5 = 30.
$$[/tex]
2. Group and combine the like terms (terms with the same power of [tex]$x$[/tex]):
- For the [tex]$x^3$[/tex] term:
[tex]$$
28x^3.
$$[/tex]
- For the [tex]$x^2$[/tex] term, add [tex]$20x^2$[/tex] and [tex]$28x^2$[/tex]:
[tex]$$
20x^2 + 28x^2 = 48x^2.
$$[/tex]
- For the [tex]$x$[/tex] term, add [tex]$20x$[/tex] and [tex]$42x$[/tex]:
[tex]$$
20x + 42x = 62x.
$$[/tex]
- The constant term remains:
[tex]$$
30.
$$[/tex]
3. Write the final resulting polynomial:
[tex]$$
28x^3 + 48x^2 + 62x + 30.
$$[/tex]
Thus, the correct answer is:
[tex]$$
\boxed{28x^3 + 48x^2 + 62x + 30}.
$$[/tex]
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