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------------------------------------------------ What is the product of [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 of [tex]$( -2x - 9y^2 )(-4x - 3)$[/tex], we can use the distributive property (also known as the FOIL method for binomials).

Here's how it works:

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 the terms:
- The expanded expression is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

Therefore, the product is [tex]$\boxed{8x^2 + 6x + 36xy^2 + 27y^2}$[/tex].