Answer :
- Multiply each term of the first polynomial by each term of the second polynomial.
- Distribute each term and simplify the expression.
- Combine like terms to obtain the final polynomial.
- The product of $(2x^2 + 3x - 1)$ and $(3x + 5)$ is $\boxed{6 x^3+19 x^2+12 x-5}$
### Explanation
1. Understanding the Problem
We are given two polynomials, $(2x^2 + 3x - 1)$ and $(3x + 5)$, and we need to find their product.
2. Plan of Action
To find the product, we will multiply each term of the first polynomial by each term of the second polynomial.
3. Multiplying the Polynomials
Let's multiply the polynomials:
$(2x^2 + 3x - 1)(3x + 5) = 2x^2(3x + 5) + 3x(3x + 5) - 1(3x + 5)$
4. Distributing the Terms
Now, distribute each term:
$= (6x^3 + 10x^2) + (9x^2 + 15x) + (-3x - 5)$
$= 6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5$
5. Combining Like Terms
Combine like terms:
$= 6x^3 + (10x^2 + 9x^2) + (15x - 3x) - 5$
$= 6x^3 + 19x^2 + 12x - 5$
6. Final Answer
The product of the two polynomials is $6x^3 + 19x^2 + 12x - 5$. Comparing this with the given options, we see that it matches option B.
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomials to model the load and stress distribution. Multiplying these polynomials helps them understand the combined effect of different factors and ensure the bridge's stability.
- Distribute each term and simplify the expression.
- Combine like terms to obtain the final polynomial.
- The product of $(2x^2 + 3x - 1)$ and $(3x + 5)$ is $\boxed{6 x^3+19 x^2+12 x-5}$
### Explanation
1. Understanding the Problem
We are given two polynomials, $(2x^2 + 3x - 1)$ and $(3x + 5)$, and we need to find their product.
2. Plan of Action
To find the product, we will multiply each term of the first polynomial by each term of the second polynomial.
3. Multiplying the Polynomials
Let's multiply the polynomials:
$(2x^2 + 3x - 1)(3x + 5) = 2x^2(3x + 5) + 3x(3x + 5) - 1(3x + 5)$
4. Distributing the Terms
Now, distribute each term:
$= (6x^3 + 10x^2) + (9x^2 + 15x) + (-3x - 5)$
$= 6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5$
5. Combining Like Terms
Combine like terms:
$= 6x^3 + (10x^2 + 9x^2) + (15x - 3x) - 5$
$= 6x^3 + 19x^2 + 12x - 5$
6. Final Answer
The product of the two polynomials is $6x^3 + 19x^2 + 12x - 5$. Comparing this with the given options, we see that it matches option B.
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer science. For example, when designing a bridge, engineers use polynomials to model the load and stress distribution. Multiplying these polynomials helps them understand the combined effect of different factors and ensure the bridge's stability.
