College

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 + 10x^2 + 15x - 5[/tex]

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

Answer :

To find the product of the polynomials [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex], you can use the distributive property, sometimes called the FOIL method when dealing with binomials, to multiply each term in the first polynomial by each term in the second polynomial.

Let's break it down step-by-step:

1. Multiply the terms from [tex]\((2x^2 + 3x - 1)\)[/tex] by [tex]\(3x\)[/tex]:

- [tex]\(2x^2 \cdot 3x = 6x^3\)[/tex]
- [tex]\(3x \cdot 3x = 9x^2\)[/tex]
- [tex]\(-1 \cdot 3x = -3x\)[/tex]

2. Multiply the terms from [tex]\((2x^2 + 3x - 1)\)[/tex] by [tex]\(5\)[/tex]:

- [tex]\(2x^2 \cdot 5 = 10x^2\)[/tex]
- [tex]\(3x \cdot 5 = 15x\)[/tex]
- [tex]\(-1 \cdot 5 = -5\)[/tex]

3. Add all the results together:

Combine the terms we obtained from each step:

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

So, the product of [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex] is:

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

Therefore, the correct answer is:

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

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