Answer :
To solve the problem of finding the product of the expressions [tex]\((2x^2 + 3x - 1)\)[/tex] and [tex]\((3x + 5)\)[/tex], we'll use the distributive property to expand the expression. Here's how you can do it step-by-step:
1. Distribute each term in the first polynomial across the second polynomial:
- First, take the term [tex]\(2x^2\)[/tex] from [tex]\((2x^2 + 3x - 1)\)[/tex] and distribute it across each term in [tex]\((3x + 5)\)[/tex]:
[tex]\[
2x^2 \times (3x + 5) = 2x^2 \times 3x + 2x^2 \times 5 = 6x^3 + 10x^2
\][/tex]
- Next, take the term [tex]\(3x\)[/tex] from [tex]\((2x^2 + 3x - 1)\)[/tex] and distribute it:
[tex]\[
3x \times (3x + 5) = 3x \times 3x + 3x \times 5 = 9x^2 + 15x
\][/tex]
- Finally, take the constant term [tex]\(-1\)[/tex] and distribute it:
[tex]\[
-1 \times (3x + 5) = -1 \times 3x - 1 \times 5 = -3x - 5
\][/tex]
2. Combine all the resulting terms:
Let's add all the terms we obtained from the distribution:
[tex]\[
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5
\][/tex]
3. Combine like terms:
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(15x - 3x = 12x\)[/tex]
The expression now becomes:
[tex]\[
6x^3 + 19x^2 + 12x - 5
\][/tex]
So, the product of the two polynomials is [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
Therefore, the correct answer is A. [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
1. Distribute each term in the first polynomial across the second polynomial:
- First, take the term [tex]\(2x^2\)[/tex] from [tex]\((2x^2 + 3x - 1)\)[/tex] and distribute it across each term in [tex]\((3x + 5)\)[/tex]:
[tex]\[
2x^2 \times (3x + 5) = 2x^2 \times 3x + 2x^2 \times 5 = 6x^3 + 10x^2
\][/tex]
- Next, take the term [tex]\(3x\)[/tex] from [tex]\((2x^2 + 3x - 1)\)[/tex] and distribute it:
[tex]\[
3x \times (3x + 5) = 3x \times 3x + 3x \times 5 = 9x^2 + 15x
\][/tex]
- Finally, take the constant term [tex]\(-1\)[/tex] and distribute it:
[tex]\[
-1 \times (3x + 5) = -1 \times 3x - 1 \times 5 = -3x - 5
\][/tex]
2. Combine all the resulting terms:
Let's add all the terms we obtained from the distribution:
[tex]\[
6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5
\][/tex]
3. Combine like terms:
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(10x^2 + 9x^2 = 19x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(15x - 3x = 12x\)[/tex]
The expression now becomes:
[tex]\[
6x^3 + 19x^2 + 12x - 5
\][/tex]
So, the product of the two polynomials is [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
Therefore, the correct answer is A. [tex]\(6x^3 + 19x^2 + 12x - 5\)[/tex].
