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Answer :
To find the product of the polynomials [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex], we will use the distributive property, also known as the FOIL method for binomials. Here's a detailed, step-by-step breakdown:
1. Distribute each term in the first polynomial to each term in the second polynomial:
First, distribute [tex]\(2x^2\)[/tex] to both [tex]\(3x\)[/tex] and [tex]\(5\)[/tex].
- [tex]\(2x^2 \times 3x = 6x^3\)[/tex]
- [tex]\(2x^2 \times 5 = 10x^2\)[/tex]
Next, distribute [tex]\(3x\)[/tex] to both [tex]\(3x\)[/tex] and [tex]\(5\)[/tex].
- [tex]\(3x \times 3x = 9x^2\)[/tex]
- [tex]\(3x \times 5 = 15x\)[/tex]
* Finally, distribute [tex]\(-1\)[/tex] to both [tex]\(3x\)[/tex] and [tex]\(5\)[/tex].
- [tex]\(-1 \times 3x = -3x\)[/tex]
- [tex]\(-1 \times 5 = -5\)[/tex]
2. Combine all the terms:
- Combine the like terms (the terms with the same powers of [tex]\(x\)[/tex]) obtained from the distribution:
- [tex]\(6x^3\)[/tex] from [tex]\(2x^2 \times 3x\)[/tex]
- [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex] from [tex]\(2x^2 \times 5\)[/tex] and [tex]\(3x \times 3x\)[/tex]
- [tex]\(15x - 3x = 12x\)[/tex] from [tex]\(3x \times 5\)[/tex] and [tex]\(-1 \times 3x\)[/tex]
- [tex]\(-5\)[/tex] from [tex]\(-1 \times 5\)[/tex]
3. Write the final expression:
The combined expression is:
- [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex]
Thus, the product of [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex] is [tex]\(\boxed{6x^3 + 19x^2 + 12x - 5}\)[/tex], which corresponds to option D.
1. Distribute each term in the first polynomial to each term in the second polynomial:
First, distribute [tex]\(2x^2\)[/tex] to both [tex]\(3x\)[/tex] and [tex]\(5\)[/tex].
- [tex]\(2x^2 \times 3x = 6x^3\)[/tex]
- [tex]\(2x^2 \times 5 = 10x^2\)[/tex]
Next, distribute [tex]\(3x\)[/tex] to both [tex]\(3x\)[/tex] and [tex]\(5\)[/tex].
- [tex]\(3x \times 3x = 9x^2\)[/tex]
- [tex]\(3x \times 5 = 15x\)[/tex]
* Finally, distribute [tex]\(-1\)[/tex] to both [tex]\(3x\)[/tex] and [tex]\(5\)[/tex].
- [tex]\(-1 \times 3x = -3x\)[/tex]
- [tex]\(-1 \times 5 = -5\)[/tex]
2. Combine all the terms:
- Combine the like terms (the terms with the same powers of [tex]\(x\)[/tex]) obtained from the distribution:
- [tex]\(6x^3\)[/tex] from [tex]\(2x^2 \times 3x\)[/tex]
- [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex] from [tex]\(2x^2 \times 5\)[/tex] and [tex]\(3x \times 3x\)[/tex]
- [tex]\(15x - 3x = 12x\)[/tex] from [tex]\(3x \times 5\)[/tex] and [tex]\(-1 \times 3x\)[/tex]
- [tex]\(-5\)[/tex] from [tex]\(-1 \times 5\)[/tex]
3. Write the final expression:
The combined expression is:
- [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex]
Thus, the product of [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex] is [tex]\(\boxed{6x^3 + 19x^2 + 12x - 5}\)[/tex], which corresponds to option D.
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