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What is the product of the expression?

[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 :

Sure! Let's find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] by using the distributive property, often called FOIL for multiplying two binomials.

### Step-by-Step Solution:

1. Multiply the first terms:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]

2. Multiply the outer terms:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]

3. Multiply the inner terms:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]

4. Multiply the last terms:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]

5. Combine all the products:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]

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

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

This matches the third option provided in the question list.

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