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Answer :
- Distribute $(3x+5)$ to each term in $(2x^2+3x-1)$.
- 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: The product is $\boxed{6 x^3+19 x^2+12 x-5}$.
### Explanation
1. Understanding the Problem
We are asked to find the product of the polynomial $(2x^2 + 3x - 1)$ and the binomial $(3x + 5)$. This involves multiplying each term of the polynomial by each term of the binomial and then combining like terms.
2. Applying the Distributive Property
To find the product, we'll use the distributive property. We multiply each term in the first expression $(2x^2 + 3x - 1)$ by each term in the second expression $(3x + 5)$. This gives us:
$(2x^2 + 3x - 1)(3x + 5) = 2x^2(3x + 5) + 3x(3x + 5) - 1(3x + 5)$
3. Distributing the Terms
Now, we distribute each term:
$2x^2(3x + 5) = 6x^3 + 10x^2$
$3x(3x + 5) = 9x^2 + 15x$
$-1(3x + 5) = -3x - 5$
4. Combining the Results
Next, we add these results together:
$(6x^3 + 10x^2) + (9x^2 + 15x) + (-3x - 5) = 6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5$
5. Combining Like Terms
Finally, we combine like terms:
$6x^3 + (10x^2 + 9x^2) + (15x - 3x) - 5 = 6x^3 + 19x^2 + 12x - 5$
6. Final Answer
The product of $(2x^2 + 3x - 1)$ and $(3x + 5)$ is $6x^3 + 19x^2 + 12x - 5$. Therefore, the correct answer is A.
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer graphics. 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, ensuring the bridge's structural integrity. Similarly, in computer graphics, polynomial multiplication is used to create complex shapes and textures by combining simpler forms.
- 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: The product is $\boxed{6 x^3+19 x^2+12 x-5}$.
### Explanation
1. Understanding the Problem
We are asked to find the product of the polynomial $(2x^2 + 3x - 1)$ and the binomial $(3x + 5)$. This involves multiplying each term of the polynomial by each term of the binomial and then combining like terms.
2. Applying the Distributive Property
To find the product, we'll use the distributive property. We multiply each term in the first expression $(2x^2 + 3x - 1)$ by each term in the second expression $(3x + 5)$. This gives us:
$(2x^2 + 3x - 1)(3x + 5) = 2x^2(3x + 5) + 3x(3x + 5) - 1(3x + 5)$
3. Distributing the Terms
Now, we distribute each term:
$2x^2(3x + 5) = 6x^3 + 10x^2$
$3x(3x + 5) = 9x^2 + 15x$
$-1(3x + 5) = -3x - 5$
4. Combining the Results
Next, we add these results together:
$(6x^3 + 10x^2) + (9x^2 + 15x) + (-3x - 5) = 6x^3 + 10x^2 + 9x^2 + 15x - 3x - 5$
5. Combining Like Terms
Finally, we combine like terms:
$6x^3 + (10x^2 + 9x^2) + (15x - 3x) - 5 = 6x^3 + 19x^2 + 12x - 5$
6. Final Answer
The product of $(2x^2 + 3x - 1)$ and $(3x + 5)$ is $6x^3 + 19x^2 + 12x - 5$. Therefore, the correct answer is A.
### Examples
Polynomial multiplication is used in various fields such as engineering, physics, and computer graphics. 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, ensuring the bridge's structural integrity. Similarly, in computer graphics, polynomial multiplication is used to create complex shapes and textures by combining simpler forms.
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