Answer :
To find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we'll use the distributive property, which involves multiplying each term in the first parenthesis by each term in the second parenthesis.
Let's go through the steps:
1. Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
This gives us the term [tex]\(8x^2\)[/tex].
2. Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
This gives us the term [tex]\(6x\)[/tex].
3. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
This gives us the term [tex]\(36xy^2\)[/tex].
4. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
This gives us the term [tex]\(27y^2\)[/tex].
Finally, we combine all these terms together to write the product:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
From the options provided, this matches:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the correct choice is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
Let's go through the steps:
1. Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
This gives us the term [tex]\(8x^2\)[/tex].
2. Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
This gives us the term [tex]\(6x\)[/tex].
3. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
This gives us the term [tex]\(36xy^2\)[/tex].
4. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
This gives us the term [tex]\(27y^2\)[/tex].
Finally, we combine all these terms together to write the product:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
From the options provided, this matches:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the correct choice is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
