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

[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]

Answer :

To find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], follow these steps:

1. Distribute Each Term in the First Parenthesis:
- Multiply [tex]\(-2x\)[/tex] by each term in the second parenthesis:
- [tex]\(-2x \times -4x = 8x^2\)[/tex]
- [tex]\(-2x \times -3 = 6x\)[/tex]

- Multiply [tex]\(-9y^2\)[/tex] by each term in the second parenthesis:
- [tex]\(-9y^2 \times -4x = 36xy^2\)[/tex]
- [tex]\(-9y^2 \times -3 = 27y^2\)[/tex]

2. Combine All Product Terms:
- Now, bring all the products together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

3. Rearrange (If Necessary):
- Order the terms typically by decreasing powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

The product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is thus:

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

This matches one of the given answer choices, which is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].