Answer :
To find the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we will use the distributive property, also known as the FOIL method (First, Outer, Inner, Last) for binomials. Here is the step-by-step solution:
1. Distribute [tex]\((-2x)\)[/tex] Across [tex]\((-4x - 3)\)[/tex]:
- First: Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x)(-4x) = 8x^2
\][/tex]
- Outer: Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x)(-3) = 6x
\][/tex]
2. Distribute [tex]\((-9y^2)\)[/tex] Across [tex]\((-4x - 3)\)[/tex]:
- Inner: Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2)(-4x) = 36xy^2
\][/tex]
- Last: Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2)(-3) = 27y^2
\][/tex]
3. Combine All Terms:
Now, add all these results together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
The correct product is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
This matches with option [tex]\( \text{option: } 8x^2 + 6x + 36xy^2 + 27y^2 \)[/tex]. So, this is the correct choice.
1. Distribute [tex]\((-2x)\)[/tex] Across [tex]\((-4x - 3)\)[/tex]:
- First: Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x)(-4x) = 8x^2
\][/tex]
- Outer: Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x)(-3) = 6x
\][/tex]
2. Distribute [tex]\((-9y^2)\)[/tex] Across [tex]\((-4x - 3)\)[/tex]:
- Inner: Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2)(-4x) = 36xy^2
\][/tex]
- Last: Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2)(-3) = 27y^2
\][/tex]
3. Combine All Terms:
Now, add all these results together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
The correct product is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
This matches with option [tex]\( \text{option: } 8x^2 + 6x + 36xy^2 + 27y^2 \)[/tex]. So, this is the correct choice.
