Answer :
The problem requires finding the product of two polynomials. The solution involves expanding the product by multiplying each term of the first polynomial by each term of the second polynomial, and then combining like terms to simplify the resulting expression. The final answer is $\boxed{2+x-31x^2-57x^3-27x^4}$.
### Explanation
1. Understanding the Problem
We are asked to find the product of two polynomials: $(1+4x+3x^2)$ and $(2-7x-9x^2)$. This means we need to multiply each term of the first polynomial by each term of the second polynomial and then combine like terms.
2. Expanding the Product
Let's expand the product:
$(1+4x+3x^2)(2-7x-9x^2) = 1(2-7x-9x^2) + 4x(2-7x-9x^2) + 3x^2(2-7x-9x^2)$
$= (2-7x-9x^2) + (8x-28x^2-36x^3) + (6x^2-21x^3-27x^4)$
3. Combining Like Terms
Now, let's combine like terms:
$= 2 + (-7x+8x) + (-9x^2-28x^2+6x^2) + (-36x^3-21x^3) + (-27x^4)$
$= 2 + x + (-37x^2+6x^2) + (-57x^3) - 27x^4$
$= 2 + x - 31x^2 - 57x^3 - 27x^4$
4. Final Answer
The product of the two polynomials is $2 + x - 31x^2 - 57x^3 - 27x^4$. Comparing this to the answer choices, 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 distribution and stress on different parts of the structure. Multiplying these polynomials helps them understand the combined effect of different factors and ensure the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces.
### Explanation
1. Understanding the Problem
We are asked to find the product of two polynomials: $(1+4x+3x^2)$ and $(2-7x-9x^2)$. This means we need to multiply each term of the first polynomial by each term of the second polynomial and then combine like terms.
2. Expanding the Product
Let's expand the product:
$(1+4x+3x^2)(2-7x-9x^2) = 1(2-7x-9x^2) + 4x(2-7x-9x^2) + 3x^2(2-7x-9x^2)$
$= (2-7x-9x^2) + (8x-28x^2-36x^3) + (6x^2-21x^3-27x^4)$
3. Combining Like Terms
Now, let's combine like terms:
$= 2 + (-7x+8x) + (-9x^2-28x^2+6x^2) + (-36x^3-21x^3) + (-27x^4)$
$= 2 + x + (-37x^2+6x^2) + (-57x^3) - 27x^4$
$= 2 + x - 31x^2 - 57x^3 - 27x^4$
4. Final Answer
The product of the two polynomials is $2 + x - 31x^2 - 57x^3 - 27x^4$. Comparing this to the answer choices, 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 distribution and stress on different parts of the structure. Multiplying these polynomials helps them understand the combined effect of different factors and ensure the bridge's stability. Similarly, in computer graphics, polynomial multiplication is used to create smooth curves and surfaces.
