High School

What is the product of [tex]\left(2x^2 + 3x - 1\right)[/tex] and [tex](3x + 5)[/tex]?

A. [tex]6x^3 + 19x^2 + 12x - 5[/tex]

B. [tex]6x^3 + 9x^2 - 3x - 5[/tex]

C. [tex]6x^3 + 19x^2 - 12x + 5[/tex]

D. [tex]6x^3 + 10x^2 + 15x - 5[/tex]

Answer :

To find the product of [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex], we need to multiply each term in the first polynomial by each term in the second polynomial. Let's do this step by step:

1. Multiply [tex]\(2x^2\)[/tex] by each term in the second polynomial:

- [tex]\(2x^2 \times 3x = 6x^3\)[/tex]
- [tex]\(2x^2 \times 5 = 10x^2\)[/tex]

2. Multiply [tex]\(3x\)[/tex] by each term in the second polynomial:

- [tex]\(3x \times 3x = 9x^2\)[/tex]
- [tex]\(3x \times 5 = 15x\)[/tex]

3. Multiply [tex]\(-1\)[/tex] by each term in the second polynomial:

- [tex]\(-1 \times 3x = -3x\)[/tex]
- [tex]\(-1 \times 5 = -5\)[/tex]

Now, combine all these results together:

[tex]\[6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5\][/tex]

Next, combine like terms:

- The [tex]\(x^3\)[/tex] term: [tex]\(6x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(15x - 3x = 12x\)[/tex]
- The constant term: [tex]\(-5\)[/tex]

So, the final expression is:

[tex]\[6x^3 + 19x^2 + 12x - 5\][/tex]

Therefore, the product is answer choice A: [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].