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------------------------------------------------ What is the product?

[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]

To find the product [tex]\((\left(-2x - 9y^2\right)(-4x - 3)\)[/tex], we need to apply the distributive property, also known as the FOIL method (First, Outer, Inner, Last) for binomials.

Let's break it down step by step:

1. First Terms:
- Multiply the first terms of each binomial: [tex]\((-2x) \times (-4x)\)[/tex]
- This equals [tex]\(8x^2\)[/tex].

2. Outer Terms:
- Multiply the outer terms: [tex]\((-2x) \times (-3)\)[/tex]
- This equals [tex]\(6x\)[/tex].

3. Inner Terms:
- Multiply the inner terms: [tex]\((-9y^2) \times (-4x)\)[/tex]
- This equals [tex]\(36xy^2\)[/tex].

4. Last Terms:
- Multiply the last terms of each binomial: [tex]\((-9y^2) \times (-3)\)[/tex]
- This equals [tex]\(27y^2\)[/tex].

Now, combine all these terms together:

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

This is the fully expanded and simplified expression for the given question. Therefore, the correct product is:

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