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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 given binomials [tex]\((-2x-9y^2)(-4x-3)\)[/tex], we use the distributive property, also known as the FOIL method when applicable to two binomials, but it works for any multiplication of polynomials.

1. Distribute each term in the first polynomial to each term in the second polynomial.

Let's go through each multiplication step by step:

- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]

- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]

- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]

- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]

2. Combine all these results together:

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

Thus, the product of the expression is:

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

This matches with the option [tex]\[8x^2 + 6x + 36xy^2 + 27y^2\][/tex] provided among the multiple choices.

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