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Answer :
Let's solve this problem step-by-step to find the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex].
1. Distribute Each Term:
We will use the distributive property to multiply each term in the first parenthesis by each term in the second parenthesis.
2. Multiply The Terms:
- First, multiply [tex]\(-2x\)[/tex] by each term in the second parenthesis:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
- Next, multiply [tex]\(-9y^2\)[/tex] by each term in the second parenthesis:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
3. Combine the Results:
Add all the results from the steps above together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So, the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
Finally, matching this expression with the given options, we find that it corresponds to:
[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]
This matches with the third option:
[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]
Therefore, the correct choice is the third option.
1. Distribute Each Term:
We will use the distributive property to multiply each term in the first parenthesis by each term in the second parenthesis.
2. Multiply The Terms:
- First, multiply [tex]\(-2x\)[/tex] by each term in the second parenthesis:
[tex]\[
(-2x) \times (-4x) = 8x^2
\][/tex]
[tex]\[
(-2x) \times (-3) = 6x
\][/tex]
- Next, multiply [tex]\(-9y^2\)[/tex] by each term in the second parenthesis:
[tex]\[
(-9y^2) \times (-4x) = 36xy^2
\][/tex]
[tex]\[
(-9y^2) \times (-3) = 27y^2
\][/tex]
3. Combine the Results:
Add all the results from the steps above together:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
So, the product of the expression [tex]\((-2x - 9y^2)(-4x - 3)\)[/tex] is [tex]\(8x^2 + 6x + 36xy^2 + 27y^2\)[/tex].
Finally, matching this expression with the given options, we find that it corresponds to:
[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]
This matches with the third option:
[tex]\[ 8x^2 + 6x + 36xy^2 + 27y^2 \][/tex]
Therefore, the correct choice is the third option.
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