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Answer :
- Distribute each term of $(3x+5)$ to $(2x^2+3x-1)$.
- Multiply: $2x^2(3x+5) + 3x(3x+5) - 1(3x+5) = 6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5$.
- Combine like terms: $6x^3 + (10x^2 + 9x^2) + (15x - 3x) - 5$.
- Simplify to get the final answer: $\boxed{6 x^3+19 x^2+12 x-5}$.
### Explanation
1. Understanding the Problem
We are asked to find the product of two polynomials: $(2x^2 + 3x - 1)$ and $(3x + 5)$. This involves multiplying each term of the first polynomial by each term of the second polynomial and then simplifying by combining like terms.
2. Distributing the Terms
To find the product, we distribute each term in the second polynomial $(3x + 5)$ to each term in the first polynomial $(2x^2 + 3x - 1)$. This gives us:
$(2x^2 + 3x - 1)(3x + 5) = 2x^2(3x + 5) + 3x(3x + 5) - 1(3x + 5)$
3. Expanding the Terms
Now, we expand each term:
$2x^2(3x + 5) = 6x^3 + 10x^2$
$3x(3x + 5) = 9x^2 + 15x$
$-1(3x + 5) = -3x - 5$
4. Combining the Expanded Terms
Next, we combine these expanded terms:
$6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5$
5. Simplifying the Expression
Finally, we simplify by combining 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 to 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, in control systems, the transfer function of a system can be represented as a rational function, which involves polynomials. Multiplying polynomials is essential for analyzing and designing these systems. Also, in computer graphics, polynomial multiplication is used in curve and surface modeling.
- Multiply: $2x^2(3x+5) + 3x(3x+5) - 1(3x+5) = 6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5$.
- Combine like terms: $6x^3 + (10x^2 + 9x^2) + (15x - 3x) - 5$.
- Simplify to get the final answer: $\boxed{6 x^3+19 x^2+12 x-5}$.
### Explanation
1. Understanding the Problem
We are asked to find the product of two polynomials: $(2x^2 + 3x - 1)$ and $(3x + 5)$. This involves multiplying each term of the first polynomial by each term of the second polynomial and then simplifying by combining like terms.
2. Distributing the Terms
To find the product, we distribute each term in the second polynomial $(3x + 5)$ to each term in the first polynomial $(2x^2 + 3x - 1)$. This gives us:
$(2x^2 + 3x - 1)(3x + 5) = 2x^2(3x + 5) + 3x(3x + 5) - 1(3x + 5)$
3. Expanding the Terms
Now, we expand each term:
$2x^2(3x + 5) = 6x^3 + 10x^2$
$3x(3x + 5) = 9x^2 + 15x$
$-1(3x + 5) = -3x - 5$
4. Combining the Expanded Terms
Next, we combine these expanded terms:
$6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5$
5. Simplifying the Expression
Finally, we simplify by combining 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 to 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, in control systems, the transfer function of a system can be represented as a rational function, which involves polynomials. Multiplying polynomials is essential for analyzing and designing these systems. Also, in computer graphics, polynomial multiplication is used in curve and surface modeling.
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