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Answer :
To find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex], we will use the distributive property. This property allows us to multiply each term in the first expression by each term in the second expression. Let's go through the process step-by-step:
1. Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
Here, the negative signs cancel each other out, and we multiply the coefficients to get 8. The [tex]\(x\)[/tex] terms combine to [tex]\(x^2\)[/tex] because we are multiplying [tex]\(x \times x\)[/tex].
2. Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
Again, the negative signs cancel, and we are left with a positive product. Multiply the coefficients [tex]\(-2\)[/tex] and [tex]\(-3\)[/tex] to get 6, and the [tex]\(x\)[/tex] remains as it is.
3. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
The negative signs cancel each other. Multiply the coefficients [tex]\(-9\)[/tex] and [tex]\(-4\)[/tex] to get 36. We have [tex]\(x\)[/tex] with [tex]\(y^2\)[/tex], which becomes [tex]\(xy^2\)[/tex].
4. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
Again, the negative signs cancel, and multiply the coefficients [tex]\(-9\)[/tex] and [tex]\(-3\)[/tex] to get 27.
Now, combine all these results to write the complete polynomial:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So, the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
This corresponds to the option:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
1. Multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
Here, the negative signs cancel each other out, and we multiply the coefficients to get 8. The [tex]\(x\)[/tex] terms combine to [tex]\(x^2\)[/tex] because we are multiplying [tex]\(x \times x\)[/tex].
2. Multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
Again, the negative signs cancel, and we are left with a positive product. Multiply the coefficients [tex]\(-2\)[/tex] and [tex]\(-3\)[/tex] to get 6, and the [tex]\(x\)[/tex] remains as it is.
3. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
The negative signs cancel each other. Multiply the coefficients [tex]\(-9\)[/tex] and [tex]\(-4\)[/tex] to get 36. We have [tex]\(x\)[/tex] with [tex]\(y^2\)[/tex], which becomes [tex]\(xy^2\)[/tex].
4. Multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
Again, the negative signs cancel, and multiply the coefficients [tex]\(-9\)[/tex] and [tex]\(-3\)[/tex] to get 27.
Now, combine all these results to write the complete polynomial:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So, the product of [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
This corresponds to the option:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
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