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

[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 solve the problem, we need to find the product of the two expressions: [tex]$(-2x - 9y^2)$[/tex] and [tex]$(-4x - 3)$[/tex].

### Step-by-step solution:

1. Multiply each term in the first expression by each term in the second expression.

Let's distribute each term separately.

- First, multiply [tex]$-2x$[/tex] by each term in [tex]$(-4x - 3)$[/tex]:
- [tex]$(-2x) \times (-4x) = 8x^2$[/tex]
This occurs because a negative times a negative gives a positive and [tex]$x \times x = x^2$[/tex].
- [tex]$(-2x) \times (-3) = 6x$[/tex]
Similarly, a negative times a negative makes a positive.

- Second, multiply [tex]$-9y^2$[/tex] by each term in [tex]$(-4x - 3)$[/tex]:
- [tex]$(-9y^2) \times (-4x) = 36xy^2$[/tex]
Again, the negatives cancel to positive, and the product includes both x and [tex]$y^2$[/tex].
- [tex]$(-9y^2) \times (-3) = 27y^2$[/tex]
Like before, the negatives make a positive.

2. Combine all the results from the multiplications:
- The complete expression after multiplication is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

3. Matching with the given options:
- The correct product is: [tex]$8x^2 + 6x + 36xy^2 + 27y^2$[/tex].

Therefore, the solution matches the choice:
[tex]\[ \text{Option: } 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]