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Find the product of the expression:

[tex]6a x^2 \left(-10a x^5 - 9x^4 - 3a \right)[/tex]

A. [tex]-60 a^2 x^7 + 54 a x^6 + 18 a^2 x^2[/tex]

B. [tex]-60 a^2 x^{10} - 54 a x^8 - 18 a^2 x^2[/tex]

C. [tex]-60 a x^7 - 54 a x^6 - 18 a x^2[/tex]

D. [tex]-60 a^2 x^7 - 54 a x^6 - 18 a^2 x^2[/tex]

Answer :

We start with the expression

[tex]$$
6a x^2 \left(-10a x^5 - 9x^4 - 3a\right).
$$[/tex]

Step 1. Multiply [tex]$6a x^2$[/tex] by each term inside the parentheses.

1. Multiply by [tex]$-10a x^5$[/tex]:

[tex]$$
6a x^2 \cdot (-10a x^5) = -60a^2 x^{2+5} = -60a^2 x^7.
$$[/tex]

2. Multiply by [tex]$-9x^4$[/tex]:

[tex]$$
6a x^2 \cdot (-9x^4) = -54a x^{2+4} = -54a x^6.
$$[/tex]

3. Multiply by [tex]$-3a$[/tex]:

[tex]$$
6a x^2 \cdot (-3a) = -18a^2 x^2.
$$[/tex]

Step 2. Combine the results:

[tex]$$
-60a^2 x^7 - 54a x^6 - 18a^2 x^2.
$$[/tex]

Thus, the product is

[tex]$$
-60a^2 x^7 - 54a x^6 - 18a^2 x^2,
$$[/tex]

which matches option D.

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