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What is the product of the expression?

[tex]\left(-2x - 9y^2\right)(-4x - 3)[/tex]

A. [tex]-8x^2 - 6x - 36xy^2 - 27y^2[/tex]

B. [tex]-14x^2 - 36xy^2 + 27y^2[/tex]

C. [tex]8x^2 + 6x + 36xy^2 + 27y^2[/tex]

D. [tex]14x^2 + 36xy^2 + 27y^2[/tex]

Answer :

To find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we'll use the distributive property, which involves multiplying each term in the first parenthesis by each term in the second parenthesis.

Let's go through the steps:

1. Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
This gives us the term [tex]\(8x^2\)[/tex].

2. Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
This gives us the term [tex]\(6x\)[/tex].

3. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
This gives us the term [tex]\(36xy^2\)[/tex].

4. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
This gives us the term [tex]\(27y^2\)[/tex].

Finally, we combine all these terms together to write the product:

[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

From the options provided, this matches:

[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

Therefore, the correct choice is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].

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