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------------------------------------------------ What is the product of the following expression?

[tex](-2x - 9y^2)(-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]

To find the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we need to use the distributive property, which allows us to multiply each term in the first parentheses by each term in the second parentheses. Let's do this step-by-step:

1. Distribute [tex]\(-2x\)[/tex] to each term in [tex]\((-4x - 3)\)[/tex]:

- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]

- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]

2. Distribute [tex]\(-9y^2\)[/tex] to each term in [tex]\((-4x - 3)\)[/tex]:

- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]

- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]

3. Add all the results together:

Combine all the terms from the previous steps:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

So, the product of [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 36xy^2 + 6x + 27y^2
\][/tex]

This matches the option [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex], which is the correct answer.