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Answer :
To solve the problem, we need to find the product of the two expressions: [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex].
### Step-by-step solution:
1. Multiply each term in the first expression by each term in the second expression.
Let's distribute each term separately.
- First, multiply [tex]\(-2x\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
- [tex]\((-2x) \times (-4x) = 8x^2\)[/tex]
This occurs because a negative times a negative gives a positive and [tex]\(x \times x = x^2\)[/tex].
- [tex]\((-2x) \times (-3) = 6x\)[/tex]
Similarly, a negative times a negative makes a positive.
- Second, multiply [tex]\(-9y^2\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
- [tex]\((-9y^2) \times (-4x) = 36xy^2\)[/tex]
Again, the negatives cancel to positive, and the product includes both x and [tex]\(y^2\)[/tex].
- [tex]\((-9y^2) \times (-3) = 27y^2\)[/tex]
Like before, the negatives make a positive.
2. Combine all the results from the multiplications:
- The complete expression after multiplication is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
3. Matching with the given options:
- The correct product is: [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
Therefore, the solution matches the choice:
[tex]\[ \text{Option: } 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]
### Step-by-step solution:
1. Multiply each term in the first expression by each term in the second expression.
Let's distribute each term separately.
- First, multiply [tex]\(-2x\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
- [tex]\((-2x) \times (-4x) = 8x^2\)[/tex]
This occurs because a negative times a negative gives a positive and [tex]\(x \times x = x^2\)[/tex].
- [tex]\((-2x) \times (-3) = 6x\)[/tex]
Similarly, a negative times a negative makes a positive.
- Second, multiply [tex]\(-9y^2\)[/tex] by each term in [tex]\((-4x - 3)\)[/tex]:
- [tex]\((-9y^2) \times (-4x) = 36xy^2\)[/tex]
Again, the negatives cancel to positive, and the product includes both x and [tex]\(y^2\)[/tex].
- [tex]\((-9y^2) \times (-3) = 27y^2\)[/tex]
Like before, the negatives make a positive.
2. Combine all the results from the multiplications:
- The complete expression after multiplication is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
3. Matching with the given options:
- The correct product is: [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
Therefore, the solution matches the choice:
[tex]\[ \text{Option: } 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]
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