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Answer :
To find the product [tex]\(\left(-2x - 9y^2\right)(-4x - 3)\)[/tex], we will use the distributive property, which involves multiplying each term in the first expression by each term in the second expression.
Let's break it down step-by-step:
1. Multiply [tex]\(-2x\)[/tex] by each term in [tex]\(-4x - 3\)[/tex]:
- [tex]\(-2x \times -4x = 8x^2\)[/tex] (Multiplying a negative number by a negative number gives a positive product, and [tex]\(x \times x = x^2\)[/tex])
- [tex]\(-2x \times -3 = 6x\)[/tex]
2. Multiply [tex]\(-9y^2\)[/tex] by each term in [tex]\(-4x - 3\)[/tex]:
- [tex]\(-9y^2 \times -4x = 36xy^2\)[/tex] (The negative signs cancel out, making the product positive.)
- [tex]\(-9y^2 \times -3 = 27y^2\)[/tex] (The negative signs cancel out, making the product positive.)
3. Combine all these results to form the expanded expression:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
After expanding, the correct product is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
By matching this expanded expression with the given multiple-choice options, we identify that the correct choice is:
[tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex]
Thus, this product matches the third option in the list provided.
Let's break it down step-by-step:
1. Multiply [tex]\(-2x\)[/tex] by each term in [tex]\(-4x - 3\)[/tex]:
- [tex]\(-2x \times -4x = 8x^2\)[/tex] (Multiplying a negative number by a negative number gives a positive product, and [tex]\(x \times x = x^2\)[/tex])
- [tex]\(-2x \times -3 = 6x\)[/tex]
2. Multiply [tex]\(-9y^2\)[/tex] by each term in [tex]\(-4x - 3\)[/tex]:
- [tex]\(-9y^2 \times -4x = 36xy^2\)[/tex] (The negative signs cancel out, making the product positive.)
- [tex]\(-9y^2 \times -3 = 27y^2\)[/tex] (The negative signs cancel out, making the product positive.)
3. Combine all these results to form the expanded expression:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
After expanding, the correct product is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
By matching this expanded expression with the given multiple-choice options, we identify that the correct choice is:
[tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex]
Thus, this product matches the third option in the list provided.
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