Answer :
To solve the problem and find the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we will apply the distributive property, often known as the FOIL method for binomials. Let's go through this step by step:
1. Distribute each term from the first binomial to each term in the second binomial.
2. Distribute [tex]\(-2x\)[/tex]:
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \cdot (-4x) = 8x^2
\][/tex]
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \cdot (-3) = 6x
\][/tex]
3. Distribute [tex]\(-9y^2\)[/tex]:
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \cdot (-4x) = 36xy^2
\][/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \cdot (-3) = 27y^2
\][/tex]
4. Combine all terms:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So the expanded expression and the product is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]
1. Distribute each term from the first binomial to each term in the second binomial.
2. Distribute [tex]\(-2x\)[/tex]:
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \cdot (-4x) = 8x^2
\][/tex]
- Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \cdot (-3) = 6x
\][/tex]
3. Distribute [tex]\(-9y^2\)[/tex]:
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \cdot (-4x) = 36xy^2
\][/tex]
- Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \cdot (-3) = 27y^2
\][/tex]
4. Combine all terms:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So the expanded expression and the product is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]
