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Answer :
To multiply the polynomials
[tex]$$
(8x^2 + 6x + 8)(6x - 5),
$$[/tex]
we use the distributive property (also known as the FOIL method for binomials, extended here).
Step 1. Multiply each term in the first polynomial by the first term in the second polynomial, [tex]$6x$[/tex]:
- Multiply [tex]$8x^2$[/tex] by [tex]$6x$[/tex]:
[tex]$$
8x^2 \cdot 6x = 48x^3
$$[/tex]
- Multiply [tex]$6x$[/tex] by [tex]$6x$[/tex]:
[tex]$$
6x \cdot 6x = 36x^2
$$[/tex]
- Multiply [tex]$8$[/tex] by [tex]$6x$[/tex]:
[tex]$$
8 \cdot 6x = 48x
$$[/tex]
So, after this step, we have:
[tex]$$
48x^3 + 36x^2 + 48x.
$$[/tex]
Step 2. Multiply each term in the first polynomial by the second term in the second polynomial, [tex]$-5$[/tex]:
- Multiply [tex]$8x^2$[/tex] by [tex]$-5$[/tex]:
[tex]$$
8x^2 \cdot (-5) = -40x^2
$$[/tex]
- Multiply [tex]$6x$[/tex] by [tex]$-5$[/tex]:
[tex]$$
6x \cdot (-5) = -30x
$$[/tex]
- Multiply [tex]$8$[/tex] by [tex]$-5$[/tex]:
[tex]$$
8 \cdot (-5) = -40
$$[/tex]
Thus, we get:
[tex]$$
-40x^2 - 30x - 40.
$$[/tex]
Step 3. Combine like terms:
Add the results from the two steps:
[tex]$$
48x^3 + 36x^2 + 48x \quad \text{and} \quad -40x^2 - 30x - 40.
$$[/tex]
- The [tex]$x^3$[/tex] term remains:
[tex]$$
48x^3.
$$[/tex]
- Combine the [tex]$x^2$[/tex] terms:
[tex]$$
36x^2 - 40x^2 = -4x^2.
$$[/tex]
- Combine the [tex]$x$[/tex] terms:
[tex]$$
48x - 30x = 18x.
$$[/tex]
- The constant term is:
[tex]$$
-40.
$$[/tex]
Thus, the final expression is:
[tex]$$
48x^3 - 4x^2 + 18x - 40.
$$[/tex]
Conclusion:
The product of the polynomials is
[tex]$$
48x^3 - 4x^2 + 18x - 40.
$$[/tex]
So, the correct answer is option A.
[tex]$$
(8x^2 + 6x + 8)(6x - 5),
$$[/tex]
we use the distributive property (also known as the FOIL method for binomials, extended here).
Step 1. Multiply each term in the first polynomial by the first term in the second polynomial, [tex]$6x$[/tex]:
- Multiply [tex]$8x^2$[/tex] by [tex]$6x$[/tex]:
[tex]$$
8x^2 \cdot 6x = 48x^3
$$[/tex]
- Multiply [tex]$6x$[/tex] by [tex]$6x$[/tex]:
[tex]$$
6x \cdot 6x = 36x^2
$$[/tex]
- Multiply [tex]$8$[/tex] by [tex]$6x$[/tex]:
[tex]$$
8 \cdot 6x = 48x
$$[/tex]
So, after this step, we have:
[tex]$$
48x^3 + 36x^2 + 48x.
$$[/tex]
Step 2. Multiply each term in the first polynomial by the second term in the second polynomial, [tex]$-5$[/tex]:
- Multiply [tex]$8x^2$[/tex] by [tex]$-5$[/tex]:
[tex]$$
8x^2 \cdot (-5) = -40x^2
$$[/tex]
- Multiply [tex]$6x$[/tex] by [tex]$-5$[/tex]:
[tex]$$
6x \cdot (-5) = -30x
$$[/tex]
- Multiply [tex]$8$[/tex] by [tex]$-5$[/tex]:
[tex]$$
8 \cdot (-5) = -40
$$[/tex]
Thus, we get:
[tex]$$
-40x^2 - 30x - 40.
$$[/tex]
Step 3. Combine like terms:
Add the results from the two steps:
[tex]$$
48x^3 + 36x^2 + 48x \quad \text{and} \quad -40x^2 - 30x - 40.
$$[/tex]
- The [tex]$x^3$[/tex] term remains:
[tex]$$
48x^3.
$$[/tex]
- Combine the [tex]$x^2$[/tex] terms:
[tex]$$
36x^2 - 40x^2 = -4x^2.
$$[/tex]
- Combine the [tex]$x$[/tex] terms:
[tex]$$
48x - 30x = 18x.
$$[/tex]
- The constant term is:
[tex]$$
-40.
$$[/tex]
Thus, the final expression is:
[tex]$$
48x^3 - 4x^2 + 18x - 40.
$$[/tex]
Conclusion:
The product of the polynomials is
[tex]$$
48x^3 - 4x^2 + 18x - 40.
$$[/tex]
So, the correct answer is option A.
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