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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]

Answer :

To find the product [tex]\(\left(-2x-9y^2\right)(-4x-3)\)[/tex], we need to use the distributive property, often referred to as the "FOIL" method for binomials.

Let's break it down step-by-step:

1. Distribute [tex]\(-2x\)[/tex]:
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \cdot (-4x) = 8x^2
\][/tex]
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \cdot (-3) = 6x
\][/tex]

2. Distribute [tex]\(-9y^2\)[/tex]:
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \cdot (-4x) = 36xy^2
\][/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \cdot (-3) = 27y^2
\][/tex]

3. Combine all terms:
- Add all the results from the distribution:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

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

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