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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 the expressions [tex]$(-2x - 9y^2)$[/tex] and [tex]$(-4x - 3)$[/tex], we need to distribute each term in the first expression by each term in the second expression and then simplify the result.

Let's break it down step by step:

1. Distribute [tex]$-2x$[/tex] to both terms 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 both terms 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. Combine all the terms we found:
- [tex]$8x^2$[/tex] from step 1
- [tex]$6x$[/tex] from step 1
- [tex]$36xy^2$[/tex] from step 2
- [tex]$27y^2$[/tex] from step 2

4. Write down the final expression:
- The expression is [tex]$8x^2 + 36xy^2 + 6x + 27y^2$[/tex]

This is the simplified product of [tex]$(-2x - 9y^2)$[/tex] and [tex]$(-4x - 3)$[/tex].