Answer :
To find the product of [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex], we'll use the distributive property, also known as the FOIL method (First, Outer, Inner, Last) for each term of the polynomials:
1. First, distribute [tex]\(3x\)[/tex] to each term in [tex]\((2x^2 + 3x - 1)\)[/tex]:
- [tex]\(3x \times 2x^2 = 6x^3\)[/tex]
- [tex]\(3x \times 3x = 9x^2\)[/tex]
- [tex]\(3x \times (-1) = -3x\)[/tex]
So, distributing [tex]\(3x\)[/tex] gives us [tex]\(6x^3 + 9x^2 - 3x\)[/tex].
2. Next, distribute [tex]\(5\)[/tex] to each term in [tex]\((2x^2 + 3x - 1)\)[/tex]:
- [tex]\(5 \times 2x^2 = 10x^2\)[/tex]
- [tex]\(5 \times 3x = 15x\)[/tex]
- [tex]\(5 \times (-1) = -5\)[/tex]
So, distributing [tex]\(5\)[/tex] gives us [tex]\(10x^2 + 15x - 5\)[/tex].
3. Now, combine all the terms from both distributions:
- [tex]\(6x^3 + 9x^2 - 3x\)[/tex]
- [tex]\(+ 10x^2 + 15x - 5\)[/tex]
4. Add the coefficients of the like terms:
- For [tex]\(x^3\)[/tex], we only have [tex]\(6x^3\)[/tex].
- For [tex]\(x^2\)[/tex], combine [tex]\(9x^2 + 10x^2 = 19x^2\)[/tex].
- For [tex]\(x\)[/tex], combine [tex]\(-3x + 15x = 12x\)[/tex].
- Lastly, the constant term is [tex]\(-5\)[/tex].
Putting it all together, the product is:
[tex]\[6x^3 + 19x^2 + 12x - 5\][/tex]
So the correct answer is:
D. [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex]
1. First, distribute [tex]\(3x\)[/tex] to each term in [tex]\((2x^2 + 3x - 1)\)[/tex]:
- [tex]\(3x \times 2x^2 = 6x^3\)[/tex]
- [tex]\(3x \times 3x = 9x^2\)[/tex]
- [tex]\(3x \times (-1) = -3x\)[/tex]
So, distributing [tex]\(3x\)[/tex] gives us [tex]\(6x^3 + 9x^2 - 3x\)[/tex].
2. Next, distribute [tex]\(5\)[/tex] to each term in [tex]\((2x^2 + 3x - 1)\)[/tex]:
- [tex]\(5 \times 2x^2 = 10x^2\)[/tex]
- [tex]\(5 \times 3x = 15x\)[/tex]
- [tex]\(5 \times (-1) = -5\)[/tex]
So, distributing [tex]\(5\)[/tex] gives us [tex]\(10x^2 + 15x - 5\)[/tex].
3. Now, combine all the terms from both distributions:
- [tex]\(6x^3 + 9x^2 - 3x\)[/tex]
- [tex]\(+ 10x^2 + 15x - 5\)[/tex]
4. Add the coefficients of the like terms:
- For [tex]\(x^3\)[/tex], we only have [tex]\(6x^3\)[/tex].
- For [tex]\(x^2\)[/tex], combine [tex]\(9x^2 + 10x^2 = 19x^2\)[/tex].
- For [tex]\(x\)[/tex], combine [tex]\(-3x + 15x = 12x\)[/tex].
- Lastly, the constant term is [tex]\(-5\)[/tex].
Putting it all together, the product is:
[tex]\[6x^3 + 19x^2 + 12x - 5\][/tex]
So the correct answer is:
D. [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex]
