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

[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]

Sure! Let's solve the problem step-by-step by expanding the given expression [tex]$(-2x - 9y^2)(-4x - 3)$[/tex].

1. Use the distributive property (FOIL method):
- Multiply each term in the first parenthesis by each term in the second parenthesis.

2. Calculate each multiplication:
- First, multiply [tex]$-2x$[/tex] by [tex]$-4x$[/tex]:
[tex]\[
(-2x)(-4x) = 8x^2
\][/tex]
- Then, multiply [tex]$-2x$[/tex] by [tex]$-3$[/tex]:
[tex]\[
(-2x)(-3) = 6x
\][/tex]
- Next, multiply [tex]$-9y^2$[/tex] by [tex]$-4x$[/tex]:
[tex]\[
(-9y^2)(-4x) = 36xy^2
\][/tex]
- Finally, multiply [tex]$-9y^2$[/tex] by [tex]$-3$[/tex]:
[tex]\[
(-9y^2)(-3) = 27y^2
\][/tex]

3. Combine all the terms together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

The product is [tex]$8x^2 + 6x + 36xy^2 + 27y^2$[/tex].

Looking at the options provided, the correct expression corresponds to:
[tex]\[ \text{Option } 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]