High School

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------------------------------------------------ What is the product of the expression?

\[ (2x - 9y^2)(-4x - 3) \]

A. \(-8x^2 - 6x - 36xy^2 - 27y^2\)
B. \(-14x^2 - 36xy^2 + 27y^2\)
C. \(8x^2 + 6x + 36xy^2 + 27y^2\)
D. \(14x^2 + 36xy^2 + 27y^2\)

Answer :

To find the product of [tex]\((2x - 9y^2) \cdot (-4x - 3)\)[/tex], we need to use the distributive property, also known as the FOIL method for binomials. Here's how it works step-by-step:

1. Distribute the first term of the first binomial to both terms of the second binomial:

[tex]\[
(2x) \cdot (-4x) = -8x^2
\][/tex]
[tex]\[
(2x) \cdot (-3) = -6x
\][/tex]

2. Distribute the second term of the first binomial to both terms of the second binomial:

[tex]\[
(-9y^2) \cdot (-4x) = 36xy^2
\][/tex]
[tex]\[
(-9y^2) \cdot (-3) = 27y^2
\][/tex]

3. Combine all the results:

The expression from all the distributions is:

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

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

Comparing this with the given options, the correct one is:

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

It looks like the signs were missed while comparing with the problem's options, so the correct calculated expression should actually be [tex]\(-8x^2 - 6x + 36xy^2 + 27y^2\)[/tex]. The easiest way could be double-checking the signs during the placement or possible multiple transcription errors.