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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 :

We want to multiply the binomials

[tex]$$
(-2x - 9y^2)(-4x - 3).
$$[/tex]

Step 1: Multiply the first term of the first binomial by each term of the second binomial:

- 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]

Step 2: Multiply the second term of the first binomial by each term of the second binomial:

- 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]

Step 3: Combine all the results:

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

Thus, the product of the two binomials is

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

This corresponds to the option:

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

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