Give me good solution What is the quotient when 3x³+4x²+8 devided by x+2
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To find the quotient when dividing \(3x^3 + 4x^2 + 8\) by \(x + 2\), we can use **polynomial long division**.

### Step-by-Step Solution:

We need to divide \( 3x^3 + 4x^2 + 0x + 8 \) by \( x + 2 \).

1. **First division**:  
   \[
   \frac{3x^3}{x} = 3x^2
   \]
   Multiply \( 3x^2 \) by \( x + 2 \), which gives:
   \[
   3x^3 + 6x^2
   \]
   Subtract this from the original polynomial:
   \[
   (3x^3 + 4x^2) - (3x^3 + 6x^2) = -2x^2
   \]

2. **Second division**:  
   \[
   \frac{-2x^2}{x} = -2x
   \]
   Multiply \( -2x \) by \( x + 2 \), which gives:
   \[
   -2x^2 - 4x
   \]
   Subtract this from the result:
   \[
   (-2x^2 + 0x) - (-2x^2 - 4x) = 4x
   \]

3. **Third division**:  
   \[
   \frac{4x}{x} = 4
   \]
   Multiply \( 4 \) by \( x + 2 \), which gives:
   \[
   4x + 8
   \]
   Subtract this from the result:
   \[
   (4x + 8) - (4x + 8) = 0
   \]

Thus, the division process ends with a remainder of 0, and the quotient is:

\[
3x^2 - 2x + 4
\]

### Final Answer:
The quotient is \( 3x^2 - 2x + 4 \).

Would you like more details or clarifications on any step?

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Here are some related questions to explore:
1. How does synthetic division work compared to polynomial long division?
2. What happens if there's a remainder in polynomial division?
3. How do you divide higher-degree polynomials by lower-degree polynomials?
4. Can the remainder theorem be applied to this problem?
5. What is the factorization of \( 3x^3 + 4x^2 + 8 \)?

**Tip**: Always organize your division work step-by-step to avoid errors.

What is the quotient when 3x³+4x²+8 divided by x+2?

Learn how to divide the polynomial 3x³ + 4x² + 8 by x + 2 using polynomial long division. This detailed solution walks through each step, breaking down the process with clear explanations. Ideal for students in Grades 10-12 studying algebra and polynomials.

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