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Answer :
Sure, let's go through the detailed step-by-step solution to multiply [tex]\((9x + 7)\)[/tex] by [tex]\( (3x^2 + 5x -1)\)[/tex].
We can use the distributive property (also known as the FOIL method when applied to binomials) to expand the multiplication. Here's the step-by-step breakdown:
1. Multiply each term in [tex]\(9x + 7\)[/tex] by each term in [tex]\(3x^2 + 5x - 1\)[/tex]:
[tex]\[
(9x + 7)(3x^2 + 5x - 1)
\][/tex]
2. Distribute [tex]\(9x\)[/tex] to each term in [tex]\(3x^2 + 5x - 1\)[/tex]:
[tex]\[
9x \cdot 3x^2 + 9x \cdot 5x + 9x \cdot (-1)
\][/tex]
[tex]\[
= 27x^3 + 45x^2 - 9x
\][/tex]
3. Distribute [tex]\(7\)[/tex] to each term in [tex]\(3x^2 + 5x - 1\)[/tex]:
[tex]\[
7 \cdot 3x^2 + 7 \cdot 5x + 7 \cdot (-1)
\][/tex]
[tex]\[
= 21x^2 + 35x - 7
\][/tex]
4. Combine all the terms from the two distributions:
[tex]\[
27x^3 + 45x^2 - 9x + 21x^2 + 35x - 7
\][/tex]
5. Group and combine like terms:
[tex]\[
27x^3 + (45x^2 + 21x^2) + (-9x + 35x) - 7
\][/tex]
[tex]\[
= 27x^3 + 66x^2 + 26x - 7
\][/tex]
So, the final expanded and simplified form of [tex]\((9x + 7)(3x^2 + 5x - 1)\)[/tex] is:
[tex]\[
27x^3 + 66x^2 + 26x - 7
\][/tex]
Hence, among the provided options, the correct answer is:
[tex]\[
27x^3 + 66x^2 + 26x - 7
\][/tex]
We can use the distributive property (also known as the FOIL method when applied to binomials) to expand the multiplication. Here's the step-by-step breakdown:
1. Multiply each term in [tex]\(9x + 7\)[/tex] by each term in [tex]\(3x^2 + 5x - 1\)[/tex]:
[tex]\[
(9x + 7)(3x^2 + 5x - 1)
\][/tex]
2. Distribute [tex]\(9x\)[/tex] to each term in [tex]\(3x^2 + 5x - 1\)[/tex]:
[tex]\[
9x \cdot 3x^2 + 9x \cdot 5x + 9x \cdot (-1)
\][/tex]
[tex]\[
= 27x^3 + 45x^2 - 9x
\][/tex]
3. Distribute [tex]\(7\)[/tex] to each term in [tex]\(3x^2 + 5x - 1\)[/tex]:
[tex]\[
7 \cdot 3x^2 + 7 \cdot 5x + 7 \cdot (-1)
\][/tex]
[tex]\[
= 21x^2 + 35x - 7
\][/tex]
4. Combine all the terms from the two distributions:
[tex]\[
27x^3 + 45x^2 - 9x + 21x^2 + 35x - 7
\][/tex]
5. Group and combine like terms:
[tex]\[
27x^3 + (45x^2 + 21x^2) + (-9x + 35x) - 7
\][/tex]
[tex]\[
= 27x^3 + 66x^2 + 26x - 7
\][/tex]
So, the final expanded and simplified form of [tex]\((9x + 7)(3x^2 + 5x - 1)\)[/tex] is:
[tex]\[
27x^3 + 66x^2 + 26x - 7
\][/tex]
Hence, among the provided options, the correct answer is:
[tex]\[
27x^3 + 66x^2 + 26x - 7
\][/tex]
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