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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 :

Sure! Let's solve the given problem step by step.

We are asked to find the product of the two expressions: [tex]\(\left(-2x - 9y^2\right)(-4x - 3)\)[/tex].

To do this, we'll use the distributive property (also known as the FOIL method for binomials):

[tex]\[
(a + b)(c + d) = a \cdot c + a \cdot d + b \cdot c + b \cdot d
\][/tex]

Here are the steps:

1. Distribute [tex]\(-2x\)[/tex] from the first expression to each term in the second expression:
[tex]\[
(-2x) \cdot (-4x) = 8x^2
\][/tex]
[tex]\[
(-2x) \cdot (-3) = 6x
\][/tex]

2. Distribute [tex]\(-9y^2\)[/tex] from the first expression to 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 we've computed:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

So, the product of [tex]\(\left(-2x - 9y^2\right)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

Matching this with the options provided:

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

So, the correct answer is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

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