Answer :
To find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we will apply the distributive property. Here's the detailed, step-by-step solution:
1. Apply the Distributive Property:
[tex]\[
(a + b)(c + d) = ac + ad + bc + bd
\][/tex]
In our expression, the terms to distribute are [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex].
2. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
3. Combine all the terms:
- After distributing, add all the products together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the expanded product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
This matches with the third option provided in the choices:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
1. Apply the Distributive Property:
[tex]\[
(a + b)(c + d) = ac + ad + bc + bd
\][/tex]
In our expression, the terms to distribute are [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex].
2. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
3. Combine all the terms:
- After distributing, add all the products together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the expanded product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
This matches with the third option provided in the choices:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
