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Answer :
- Distribute $2a$ in $2a(a^2 + 4a + 6)$ to get $2a^3 + 8a^2 + 12a$.
- Divide each term in $(m^4 + m^3 + 4m^2)$ by $m^2$ to get $m^2 + m + 4$.
- Distribute $-5x$ in $-5x(7x^4 - 9x^2 + 1)$ to get $-35x^5 + 45x^3 - 5x$.
- The answers are $2 a^3+8 a^2+12 a$, $m^2+m+4$, and $-35 x^5+45 x^3-5 x$.
### Explanation
1. Understanding the Problem
We need to find the correct algebraic expression for each of the given operations.
2. Solving part a
a. To find the product of $2a(a^2 + 4a + 6)$, we distribute $2a$ to each term inside the parenthesis: $2a * a^2 + 2a * 4a + 2a * 6$. This simplifies to $2a^3 + 8a^2 + 12a$.
3. Solving part b
b. To find the quotient of $(m^4 + m^3 + 4m^2) \\div m^2$, we divide each term in the numerator by $m^2$: $m^4/m^2 + m^3/m^2 + 4m^2/m^2$. This simplifies to $m^2 + m + 4$.
4. Solving part c
c. To find the product of $-5x(7x^4 - 9x^2 + 1)$, we distribute $-5x$ to each term inside the parenthesis: $-5x * 7x^4 + (-5x) * (-9x^2) + (-5x) * 1$. This simplifies to $-35x^5 + 45x^3 - 5x$.
5. Final Answer
Therefore, the correct answers are:
a. $2 a^3+8 a^2+12 a$
b. $m^2+m+4$
c. $-35 x^5+45 x^3-5 x$
### Examples
Understanding polynomial multiplication and division is crucial in various fields, such as engineering, computer science, and economics. For instance, engineers use polynomial operations to model and analyze systems, while economists use them to create models for economic growth and prediction. In computer graphics, these operations are fundamental for rendering complex shapes and creating realistic animations. By mastering these concepts, you'll be equipped to tackle a wide range of real-world problems.
- Divide each term in $(m^4 + m^3 + 4m^2)$ by $m^2$ to get $m^2 + m + 4$.
- Distribute $-5x$ in $-5x(7x^4 - 9x^2 + 1)$ to get $-35x^5 + 45x^3 - 5x$.
- The answers are $2 a^3+8 a^2+12 a$, $m^2+m+4$, and $-35 x^5+45 x^3-5 x$.
### Explanation
1. Understanding the Problem
We need to find the correct algebraic expression for each of the given operations.
2. Solving part a
a. To find the product of $2a(a^2 + 4a + 6)$, we distribute $2a$ to each term inside the parenthesis: $2a * a^2 + 2a * 4a + 2a * 6$. This simplifies to $2a^3 + 8a^2 + 12a$.
3. Solving part b
b. To find the quotient of $(m^4 + m^3 + 4m^2) \\div m^2$, we divide each term in the numerator by $m^2$: $m^4/m^2 + m^3/m^2 + 4m^2/m^2$. This simplifies to $m^2 + m + 4$.
4. Solving part c
c. To find the product of $-5x(7x^4 - 9x^2 + 1)$, we distribute $-5x$ to each term inside the parenthesis: $-5x * 7x^4 + (-5x) * (-9x^2) + (-5x) * 1$. This simplifies to $-35x^5 + 45x^3 - 5x$.
5. Final Answer
Therefore, the correct answers are:
a. $2 a^3+8 a^2+12 a$
b. $m^2+m+4$
c. $-35 x^5+45 x^3-5 x$
### Examples
Understanding polynomial multiplication and division is crucial in various fields, such as engineering, computer science, and economics. For instance, engineers use polynomial operations to model and analyze systems, while economists use them to create models for economic growth and prediction. In computer graphics, these operations are fundamental for rendering complex shapes and creating realistic animations. By mastering these concepts, you'll be equipped to tackle a wide range of real-world problems.
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