Answer :
- Distribute $(2x+6)$ over $(5x^2-x-3)$.
- Multiply each term: $5x^2(2x+6)$, $-x(2x+6)$, and $-3(2x+6)$.
- Expand and combine like terms: $10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18$.
- Simplify to get the final answer: $\boxed{10 x^3+28 x^2-12 x-18}$.
### Explanation
1. Understanding the Problem
We are given two polynomials, $(5x^2 - x - 3)$ and $(2x + 6)$, and we need to find their product. This involves multiplying each term of the first polynomial by each term of the second polynomial and then simplifying by combining like terms.
2. Applying the Distributive Property
To find the product of the polynomials $(5x^2 - x - 3)$ and $(2x + 6)$, we use the distributive property. We multiply each term in the first polynomial by each term in the second polynomial:
$(5x^2 - x - 3)(2x + 6) = 5x^2(2x + 6) - x(2x + 6) - 3(2x + 6)$
3. Expanding Each Term
Now, we expand each term:
$5x^2(2x + 6) = 10x^3 + 30x^2$
$-x(2x + 6) = -2x^2 - 6x$
$-3(2x + 6) = -6x - 18$
4. Combining the Terms
Next, we combine the expanded terms:
$10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18$
5. Simplifying the Expression
Finally, we simplify by combining like terms:
$10x^3 + (30x^2 - 2x^2) + (-6x - 6x) - 18 = 10x^3 + 28x^2 - 12x - 18$
6. Identifying the Correct Option
Comparing our result with the given options, we see that the correct answer is:
$10x^3 + 28x^2 - 12x - 18$
### 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 complex shapes and textures by combining simpler geometric forms.
- Multiply each term: $5x^2(2x+6)$, $-x(2x+6)$, and $-3(2x+6)$.
- Expand and combine like terms: $10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18$.
- Simplify to get the final answer: $\boxed{10 x^3+28 x^2-12 x-18}$.
### Explanation
1. Understanding the Problem
We are given two polynomials, $(5x^2 - x - 3)$ and $(2x + 6)$, and we need to find their product. This involves multiplying each term of the first polynomial by each term of the second polynomial and then simplifying by combining like terms.
2. Applying the Distributive Property
To find the product of the polynomials $(5x^2 - x - 3)$ and $(2x + 6)$, we use the distributive property. We multiply each term in the first polynomial by each term in the second polynomial:
$(5x^2 - x - 3)(2x + 6) = 5x^2(2x + 6) - x(2x + 6) - 3(2x + 6)$
3. Expanding Each Term
Now, we expand each term:
$5x^2(2x + 6) = 10x^3 + 30x^2$
$-x(2x + 6) = -2x^2 - 6x$
$-3(2x + 6) = -6x - 18$
4. Combining the Terms
Next, we combine the expanded terms:
$10x^3 + 30x^2 - 2x^2 - 6x - 6x - 18$
5. Simplifying the Expression
Finally, we simplify by combining like terms:
$10x^3 + (30x^2 - 2x^2) + (-6x - 6x) - 18 = 10x^3 + 28x^2 - 12x - 18$
6. Identifying the Correct Option
Comparing our result with the given options, we see that the correct answer is:
$10x^3 + 28x^2 - 12x - 18$
### 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 complex shapes and textures by combining simpler geometric forms.
