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Answer :
To multiply the polynomials
[tex]$$
(8x^2 + 6x + 8)(6x - 5),
$$[/tex]
we start by distributing each term in the first polynomial to every term in the second polynomial.
1. Multiply the first term, [tex]$8x^2$[/tex], by each term of [tex]$(6x - 5)$[/tex]:
[tex]$$
8x^2 \cdot 6x = 48x^3,\quad 8x^2 \cdot (-5) = -40x^2.
$$[/tex]
So, the contribution is:
[tex]$$
48x^3 - 40x^2.
$$[/tex]
2. Multiply the second term, [tex]$6x$[/tex], by each term of [tex]$(6x - 5)$[/tex]:
[tex]$$
6x \cdot 6x = 36x^2,\quad 6x \cdot (-5) = -30x.
$$[/tex]
So, the contribution is:
[tex]$$
36x^2 - 30x.
$$[/tex]
3. Multiply the third term, [tex]$8$[/tex], by each term of [tex]$(6x - 5)$[/tex]:
[tex]$$
8 \cdot 6x = 48x,\quad 8 \cdot (-5) = -40.
$$[/tex]
So, the contribution is:
[tex]$$
48x - 40.
$$[/tex]
4. Now, add all the parts together:
[tex]$$
\begin{aligned}
(48x^3 - 40x^2) &+ (36x^2 - 30x) + (48x - 40) \\
&= 48x^3 + (-40x^2 + 36x^2) + (-30x + 48x) - 40 \\
&= 48x^3 - 4x^2 + 18x - 40.
\end{aligned}
$$[/tex]
The final result of multiplying the polynomials is
[tex]$$
48x^3 - 4x^2 + 18x - 40.
$$[/tex]
Thus, the correct answer is:
D. [tex]$48x^3 - 4x^2 + 18x - 40$[/tex].
[tex]$$
(8x^2 + 6x + 8)(6x - 5),
$$[/tex]
we start by distributing each term in the first polynomial to every term in the second polynomial.
1. Multiply the first term, [tex]$8x^2$[/tex], by each term of [tex]$(6x - 5)$[/tex]:
[tex]$$
8x^2 \cdot 6x = 48x^3,\quad 8x^2 \cdot (-5) = -40x^2.
$$[/tex]
So, the contribution is:
[tex]$$
48x^3 - 40x^2.
$$[/tex]
2. Multiply the second term, [tex]$6x$[/tex], by each term of [tex]$(6x - 5)$[/tex]:
[tex]$$
6x \cdot 6x = 36x^2,\quad 6x \cdot (-5) = -30x.
$$[/tex]
So, the contribution is:
[tex]$$
36x^2 - 30x.
$$[/tex]
3. Multiply the third term, [tex]$8$[/tex], by each term of [tex]$(6x - 5)$[/tex]:
[tex]$$
8 \cdot 6x = 48x,\quad 8 \cdot (-5) = -40.
$$[/tex]
So, the contribution is:
[tex]$$
48x - 40.
$$[/tex]
4. Now, add all the parts together:
[tex]$$
\begin{aligned}
(48x^3 - 40x^2) &+ (36x^2 - 30x) + (48x - 40) \\
&= 48x^3 + (-40x^2 + 36x^2) + (-30x + 48x) - 40 \\
&= 48x^3 - 4x^2 + 18x - 40.
\end{aligned}
$$[/tex]
The final result of multiplying the polynomials is
[tex]$$
48x^3 - 4x^2 + 18x - 40.
$$[/tex]
Thus, the correct answer is:
D. [tex]$48x^3 - 4x^2 + 18x - 40$[/tex].
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