Answer :
To multiply the polynomials
[tex]$$
(4x^2+4x+6)(7x+5),
$$[/tex]
follow these steps:
1. Multiply each term in the first polynomial by the entire second polynomial:
- Multiply the first term [tex]$4x^2$[/tex] by [tex]$(7x+5)$[/tex]:
[tex]$$
4x^2 \cdot 7x = 28x^3 \quad \text{and} \quad 4x^2 \cdot 5 = 20x^2.
$$[/tex]
So,
[tex]$$
4x^2(7x+5) = 28x^3 + 20x^2.
$$[/tex]
2. Multiply the second term [tex]$4x$[/tex] by [tex]$(7x+5)$[/tex]:
[tex]$$
4x \cdot 7x = 28x^2 \quad \text{and} \quad 4x \cdot 5 = 20x.
$$[/tex]
Thus,
[tex]$$
4x(7x+5) = 28x^2 + 20x.
$$[/tex]
3. Multiply the third term [tex]$6$[/tex] by [tex]$(7x+5)$[/tex]:
[tex]$$
6 \cdot 7x = 42x \quad \text{and} \quad 6 \cdot 5 = 30.
$$[/tex]
Therefore,
[tex]$$
6(7x+5) = 42x + 30.
$$[/tex]
4. Add all the results together and combine like terms:
Write the sum:
[tex]$$
(28x^3 + 20x^2) + (28x^2 + 20x) + (42x + 30).
$$[/tex]
Now, combine like terms:
- The cubic term:
[tex]$$
28x^3.
$$[/tex]
- The quadratic terms:
[tex]$$
20x^2 + 28x^2 = 48x^2.
$$[/tex]
- The linear terms:
[tex]$$
20x + 42x = 62x.
$$[/tex]
- The constant term:
[tex]$$
30.
$$[/tex]
Hence, the final result is:
[tex]$$
28x^3 + 48x^2 + 62x + 30.
$$[/tex]
Comparing the result with the provided answer choices, the correct answer is:
D. [tex]$$28x^3+48x^2+62x+30.$$[/tex]
[tex]$$
(4x^2+4x+6)(7x+5),
$$[/tex]
follow these steps:
1. Multiply each term in the first polynomial by the entire second polynomial:
- Multiply the first term [tex]$4x^2$[/tex] by [tex]$(7x+5)$[/tex]:
[tex]$$
4x^2 \cdot 7x = 28x^3 \quad \text{and} \quad 4x^2 \cdot 5 = 20x^2.
$$[/tex]
So,
[tex]$$
4x^2(7x+5) = 28x^3 + 20x^2.
$$[/tex]
2. Multiply the second term [tex]$4x$[/tex] by [tex]$(7x+5)$[/tex]:
[tex]$$
4x \cdot 7x = 28x^2 \quad \text{and} \quad 4x \cdot 5 = 20x.
$$[/tex]
Thus,
[tex]$$
4x(7x+5) = 28x^2 + 20x.
$$[/tex]
3. Multiply the third term [tex]$6$[/tex] by [tex]$(7x+5)$[/tex]:
[tex]$$
6 \cdot 7x = 42x \quad \text{and} \quad 6 \cdot 5 = 30.
$$[/tex]
Therefore,
[tex]$$
6(7x+5) = 42x + 30.
$$[/tex]
4. Add all the results together and combine like terms:
Write the sum:
[tex]$$
(28x^3 + 20x^2) + (28x^2 + 20x) + (42x + 30).
$$[/tex]
Now, combine like terms:
- The cubic term:
[tex]$$
28x^3.
$$[/tex]
- The quadratic terms:
[tex]$$
20x^2 + 28x^2 = 48x^2.
$$[/tex]
- The linear terms:
[tex]$$
20x + 42x = 62x.
$$[/tex]
- The constant term:
[tex]$$
30.
$$[/tex]
Hence, the final result is:
[tex]$$
28x^3 + 48x^2 + 62x + 30.
$$[/tex]
Comparing the result with the provided answer choices, the correct answer is:
D. [tex]$$28x^3+48x^2+62x+30.$$[/tex]
