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Answer :
Certainly! Let's find the product of the polynomials [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex] step-by-step:
1. Identify the terms in each polynomial:
- The first polynomial is [tex]\( -2x - 9y^2 \)[/tex].
- Here we have terms: [tex]\( -2x \)[/tex] and [tex]\( -9y^2 \)[/tex].
- The second polynomial is [tex]\( -4x - 3 \)[/tex].
- Here we have terms: [tex]\( -4x \)[/tex] and [tex]\( -3 \)[/tex].
2. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply each term in [tex]\( (-2x - 9y^2) \)[/tex] by each term in [tex]\( (-4x - 3) \)[/tex].
3. Calculate each product individually:
- First, we multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
-2x \cdot -4x = 8x^2
\][/tex]
- Next, we multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-2x \cdot -3 = 6x
\][/tex]
- Then, we multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
-9y^2 \cdot -4x = 36xy^2
\][/tex]
- Finally, we multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-9y^2 \cdot -3 = 27y^2
\][/tex]
4. Combine all the results to form the polynomial:
- Now, we sum all the individual products:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Thus, the product of the polynomials [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex] is:
[tex]\[
\boxed{8x^2 + 6x + 36xy^2 + 27y^2}
\][/tex]
1. Identify the terms in each polynomial:
- The first polynomial is [tex]\( -2x - 9y^2 \)[/tex].
- Here we have terms: [tex]\( -2x \)[/tex] and [tex]\( -9y^2 \)[/tex].
- The second polynomial is [tex]\( -4x - 3 \)[/tex].
- Here we have terms: [tex]\( -4x \)[/tex] and [tex]\( -3 \)[/tex].
2. Distribute each term in the first polynomial to each term in the second polynomial:
- Multiply each term in [tex]\( (-2x - 9y^2) \)[/tex] by each term in [tex]\( (-4x - 3) \)[/tex].
3. Calculate each product individually:
- First, we multiply [tex]\(-2x\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
-2x \cdot -4x = 8x^2
\][/tex]
- Next, we multiply [tex]\(-2x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-2x \cdot -3 = 6x
\][/tex]
- Then, we multiply [tex]\(-9y^2\)[/tex] by [tex]\(-4x\)[/tex]:
[tex]\[
-9y^2 \cdot -4x = 36xy^2
\][/tex]
- Finally, we multiply [tex]\(-9y^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-9y^2 \cdot -3 = 27y^2
\][/tex]
4. Combine all the results to form the polynomial:
- Now, we sum all the individual products:
[tex]\[
8x^2 + 6x + 36xy^2 + 27y^2
\][/tex]
Thus, the product of the polynomials [tex]\((-2x - 9y^2)\)[/tex] and [tex]\((-4x - 3)\)[/tex] is:
[tex]\[
\boxed{8x^2 + 6x + 36xy^2 + 27y^2}
\][/tex]
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