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Answer :
Sure! To find the product of [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex], we need to use the distributive property, which means we'll distribute each term in the first polynomial to each term in the second polynomial.
Here's how you can break it down:
1. Multiply each term in the first expression [tex]\((2x^2 + 3x - 1)\)[/tex] by every term in the second expression [tex]\((3x + 5)\)[/tex].
Step-by-step calculation:
- First, distribute [tex]\(2x^2\)[/tex] to both terms in [tex]\((3x + 5)\)[/tex]:
[tex]\[
2x^2 \times 3x = 6x^3
\][/tex]
[tex]\[
2x^2 \times 5 = 10x^2
\][/tex]
- Next, distribute [tex]\(3x\)[/tex] to both terms in [tex]\((3x + 5)\)[/tex]:
[tex]\[
3x \times 3x = 9x^2
\][/tex]
[tex]\[
3x \times 5 = 15x
\][/tex]
- Then, distribute [tex]\(-1\)[/tex] to both terms in [tex]\((3x + 5)\)[/tex]:
[tex]\[
-1 \times 3x = -3x
\][/tex]
[tex]\[
-1 \times 5 = -5
\][/tex]
2. Now, add all the terms obtained from these multiplications:
[tex]\[
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(15x - 3x = 12x\)[/tex]
So, the final expanded product is:
[tex]\[
6x^3 + 19x^2 + 12x - 5
\][/tex]
Therefore, the correct answer is B. [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
Here's how you can break it down:
1. Multiply each term in the first expression [tex]\((2x^2 + 3x - 1)\)[/tex] by every term in the second expression [tex]\((3x + 5)\)[/tex].
Step-by-step calculation:
- First, distribute [tex]\(2x^2\)[/tex] to both terms in [tex]\((3x + 5)\)[/tex]:
[tex]\[
2x^2 \times 3x = 6x^3
\][/tex]
[tex]\[
2x^2 \times 5 = 10x^2
\][/tex]
- Next, distribute [tex]\(3x\)[/tex] to both terms in [tex]\((3x + 5)\)[/tex]:
[tex]\[
3x \times 3x = 9x^2
\][/tex]
[tex]\[
3x \times 5 = 15x
\][/tex]
- Then, distribute [tex]\(-1\)[/tex] to both terms in [tex]\((3x + 5)\)[/tex]:
[tex]\[
-1 \times 3x = -3x
\][/tex]
[tex]\[
-1 \times 5 = -5
\][/tex]
2. Now, add all the terms obtained from these multiplications:
[tex]\[
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(15x - 3x = 12x\)[/tex]
So, the final expanded product is:
[tex]\[
6x^3 + 19x^2 + 12x - 5
\][/tex]
Therefore, the correct answer is B. [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
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