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Answer :
To solve the problem, we need to find the product of the expressions [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex]. Let's break it down step-by-step:
1. Distribute each term in the first expression to each term in the second expression:
[tex]\[
(-2x - 9y^2)(-4x - 3)
\][/tex]
This means we will apply the distributive property to each pair of terms from the two binomials.
2. Multiply [tex]\(-2x\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
[tex]\[
-2x \cdot (-4x) + (-2x) \cdot (-3)
\][/tex]
- [tex]\(-2x \cdot (-4x) = 8x^2\)[/tex] (since negative times negative gives a positive, and [tex]\(x \cdot x = x^2\)[/tex])
- [tex]\((-2x) \cdot (-3) = 6x\)[/tex] (since negative times negative gives a positive)
3. Multiply [tex]\(-9y^2\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
[tex]\[
-9y^2 \cdot (-4x) + (-9y^2) \cdot (-3)
\][/tex]
- [tex]\(-9y^2 \cdot (-4x) = 36xy^2\)[/tex] (since negative times negative gives a positive, and [tex]\(y^2\)[/tex] is multiplied by [tex]\(x\)[/tex])
- [tex]\((-9y^2) \cdot (-3) = 27y^2\)[/tex] (since negative times negative gives a positive, and keeping the same power of [tex]\(y\)[/tex])
4. Combine all the products obtained:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the product of the expressions [tex]\(\left(-2x - 9y^2\right)\)[/tex] and [tex]\(\left(-4x - 3\right)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Hence, the correct choice from the given options is:
[tex]\[
8 x^2 + 6 x + 36 x y^2 + 27 y^2
\][/tex]
1. Distribute each term in the first expression to each term in the second expression:
[tex]\[
(-2x - 9y^2)(-4x - 3)
\][/tex]
This means we will apply the distributive property to each pair of terms from the two binomials.
2. Multiply [tex]\(-2x\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
[tex]\[
-2x \cdot (-4x) + (-2x) \cdot (-3)
\][/tex]
- [tex]\(-2x \cdot (-4x) = 8x^2\)[/tex] (since negative times negative gives a positive, and [tex]\(x \cdot x = x^2\)[/tex])
- [tex]\((-2x) \cdot (-3) = 6x\)[/tex] (since negative times negative gives a positive)
3. Multiply [tex]\(-9y^2\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
[tex]\[
-9y^2 \cdot (-4x) + (-9y^2) \cdot (-3)
\][/tex]
- [tex]\(-9y^2 \cdot (-4x) = 36xy^2\)[/tex] (since negative times negative gives a positive, and [tex]\(y^2\)[/tex] is multiplied by [tex]\(x\)[/tex])
- [tex]\((-9y^2) \cdot (-3) = 27y^2\)[/tex] (since negative times negative gives a positive, and keeping the same power of [tex]\(y\)[/tex])
4. Combine all the products obtained:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Therefore, the product of the expressions [tex]\(\left(-2x - 9y^2\right)\)[/tex] and [tex]\(\left(-4x - 3\right)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Hence, the correct choice from the given options is:
[tex]\[
8 x^2 + 6 x + 36 x y^2 + 27 y^2
\][/tex]
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