College

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

A. [tex]6x^3 + 9x^2 - 3x - 5[/tex]
B. [tex]6x^3 + 19x^2 + 12x - 5[/tex]
C. [tex]6x^3 + 19x^2 - 12x + 5[/tex]
D. [tex]6x^3 + 10x^2 + 15x - 5[/tex]

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].