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

Answer :

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

1. Multiply the First Term of [tex]\((2x^2 + 3x - 1)\)[/tex]:

[tex]\[
2x^2 \times 3x = 6x^3
\][/tex]

[tex]\[
2x^2 \times 5 = 10x^2
\][/tex]

2. Multiply the Second Term of [tex]\((2x^2 + 3x - 1)\)[/tex]:

[tex]\[
3x \times 3x = 9x^2
\][/tex]

[tex]\[
3x \times 5 = 15x
\][/tex]

3. Multiply the Third Term of [tex]\((2x^2 + 3x - 1)\)[/tex]:

[tex]\[
-1 \times 3x = -3x
\][/tex]

[tex]\[
-1 \times 5 = -5
\][/tex]

4. Combine All the Products:

Now, add all these products together:

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

5. Combine Like Terms:

- Combine [tex]\(10x^2\)[/tex] and [tex]\(9x^2\)[/tex] to get [tex]\(19x^2\)[/tex].
- Combine [tex]\(15x\)[/tex] and [tex]\(-3x\)[/tex] to get [tex]\(12x\)[/tex].

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

So the correct answer is:

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

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