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------------------------------------------------ What is the product of the 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], let's multiply these expressions using the distributive property, also known as the FOIL method when dealing with binomials.

Here's a step-by-step breakdown:

1. Multiply each term in the first binomial by each term in the second binomial:

- First, multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \cdot (-4x) = 8x^2
\][/tex]

- Next, multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \cdot (-3) = 6x
\][/tex]

- Then, multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \cdot (-4x) = 36xy^2
\][/tex]

- Finally, multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \cdot (-3) = 27y^2
\][/tex]

2. Combine all these products:

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

This gives us the final expanded expression. Therefore, the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

The answer is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].