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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 of finding the product [tex]\(( -2x - 9y^2)(-4x - 3)\)[/tex], we need to distribute each term in the first polynomial by each term in the second polynomial. Here's a step-by-step explanation:

1. First, distribute [tex]\(-2x\)[/tex] over each term in the second parenthesis [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. Next, distribute [tex]\(-9y^2\)[/tex] over each term in the second parenthesis [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 results from the distributions:

The expression becomes:
[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]

This is the expanded polynomial after multiplying the two given polynomials. Therefore, the correct product is:

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

This matches the option: [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].