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Answer :
To solve the problem [tex]\(\left(-2 x - 9 y^2\right)(-4 x - 3)\)[/tex] and find the product, follow these steps:
1. Distribute each term in the first polynomial to each term in the second polynomial:
This means we apply the distributive property, which states that [tex]\(a(b + c) = ab + ac\)[/tex], to each term within the parentheses:
[tex]\[
(-2x - 9y^2)(-4x - 3) = (-2x)(-4x) + (-2x)(-3) + (-9y^2)(-4x) + (-9y^2)(-3)
\][/tex]
2. Multiply each pair of terms:
- For [tex]\((-2x)(-4x)\)[/tex]:
[tex]\[
(-2)(-4) \cdot x \cdot x = 8x^2
\][/tex]
- For [tex]\((-2x)(-3)\)[/tex]:
[tex]\[
(-2)(-3) \cdot x = 6x
\][/tex]
- For [tex]\((-9y^2)(-4x)\)[/tex]:
[tex]\[
(-9)(-4) \cdot y^2 \cdot x = 36xy^2
\][/tex]
- For [tex]\((-9y^2)(-3)\)[/tex]:
[tex]\[
(-9)(-3) \cdot y^2 = 27y^2
\][/tex]
3. Combine all the terms together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So, the product of [tex]\(\left(-2 x - 9 y^2\right)(-4 x - 3)\)[/tex] is:
[tex]\[
\boxed{8x^2 + 6x + 36xy^2 + 27y^2}
\][/tex]
Which matches the given options. The correct choice is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
1. Distribute each term in the first polynomial to each term in the second polynomial:
This means we apply the distributive property, which states that [tex]\(a(b + c) = ab + ac\)[/tex], to each term within the parentheses:
[tex]\[
(-2x - 9y^2)(-4x - 3) = (-2x)(-4x) + (-2x)(-3) + (-9y^2)(-4x) + (-9y^2)(-3)
\][/tex]
2. Multiply each pair of terms:
- For [tex]\((-2x)(-4x)\)[/tex]:
[tex]\[
(-2)(-4) \cdot x \cdot x = 8x^2
\][/tex]
- For [tex]\((-2x)(-3)\)[/tex]:
[tex]\[
(-2)(-3) \cdot x = 6x
\][/tex]
- For [tex]\((-9y^2)(-4x)\)[/tex]:
[tex]\[
(-9)(-4) \cdot y^2 \cdot x = 36xy^2
\][/tex]
- For [tex]\((-9y^2)(-3)\)[/tex]:
[tex]\[
(-9)(-3) \cdot y^2 = 27y^2
\][/tex]
3. Combine all the terms together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So, the product of [tex]\(\left(-2 x - 9 y^2\right)(-4 x - 3)\)[/tex] is:
[tex]\[
\boxed{8x^2 + 6x + 36xy^2 + 27y^2}
\][/tex]
Which matches the given options. The correct choice is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
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