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.
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.
