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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 [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we can expand the expression using the distributive property (also known as the FOIL method for binomials).

Let's expand step by step:

1. Multiply [tex]\(-2x\)[/tex] by each term in the second expression:

[tex]\[
-2x \cdot (-4x) = 8x^2
\][/tex]

[tex]\[
-2x \cdot (-3) = 6x
\][/tex]

2. Multiply [tex]\(-9y^2\)[/tex] by each term in the second expression:

[tex]\[
-9y^2 \cdot (-4x) = 36xy^2
\][/tex]

[tex]\[
-9y^2 \cdot (-3) = 27y^2
\][/tex]

3. Combine all the terms:

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

Therefore, the expanded product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:

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

This matches one of the provided options:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

So, this is the correct answer.

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