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The image asks to use long division to divide the polynomial \( 6x^4 - 25x^3 + 8x^2 + 27x + 17 \) by \( 2x^2 - 3x - 2 \), and express the result in the form \( q(x) + \frac{r(x)}{b(x)} \), where \( q(x) \) is the quotient, \( r(x) \) is the remainder, and \( b(x) \) is the divisor.

Let's go through the long division process.

### Step 1: Set up the division
We want to divide:
\[
\frac{6x^4 - 25x^3 + 8x^2 + 27x + 17}{2x^2 - 3x - 2}
\]

### Step 2: Perform long division

1. **First Division**: Divide the leading term \( 6x^4 \) by the leading term \( 2x^2 \), which gives \( 3x^2 \). Multiply the divisor by \( 3x^2 \):
   \[
   (2x^2 - 3x - 2) \cdot 3x^2 = 6x^4 - 9x^3 - 6x^2
   \]
   Subtract this from the original polynomial:
   \[
   (6x^4 - 25x^3 + 8x^2 + 27x + 17) - (6x^4 - 9x^3 - 6x^2) = -16x^3 + 14x^2 + 27x + 17
   \]

2. **Second Division**: Divide the new leading term \( -16x^3 \) by \( 2x^2 \), which gives \( -8x \). Multiply the divisor by \( -8x \):
   \[
   (2x^2 - 3x - 2) \cdot -8x = -16x^3 + 24x^2 + 16x
   \]
   Subtract this from the result of the first step:
   \[
   (-16x^3 + 14x^2 + 27x + 17) - (-16x^3 + 24x^2 + 16x) = -10x^2 + 11x + 17
   \]

3. **Third Division**: Divide the new leading term \( -10x^2 \) by \( 2x^2 \), which gives \( -5 \). Multiply the divisor by \( -5 \):
   \[
   (2x^2 - 3x - 2) \cdot -5 = -10x^2 + 15x + 10
   \]
   Subtract this from the result of the second step:
   \[
   (-10x^2 + 11x + 17) - (-10x^2 + 15x + 10) = -4x + 7
   \]

Now we have the quotient \( q(x) = 3x^2 - 8x - 5 \) and the remainder \( r(x) = -4x + 7 \).

### Final Answer:
\[
\frac{6x^4 - 25x^3 + 8x^2 + 27x + 17}{2x^2 - 3x - 2} = 3x^2 - 8x - 5 + \frac{-4x + 7}{2x^2 - 3x - 2}
\]

Would you like further details on any step, or do you have other questions?

Here are 5 questions that expand on this topic:
1. How can synthetic division simplify this process when dividing by linear polynomials?
2. What are the benefits of using long division in polynomial division compared to other methods?
3. Can this division be solved using a remainder theorem?
4. How can this result be applied in real-world problems, such as optimization?
5. What happens when dividing two polynomials of the same degree?

**Tip:** When dividing polynomials, always ensure the terms are written in descending order of degrees for accurate calculations.

Use the long division method to find the result when 6x^4 - 25x^3 + 8x^2 + 27x + 17 is divided by 2x^2 - 3x - 2. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Solve the polynomial division of 6x^4 - 25x^3 + 8x^2 + 27x + 17 by 2x^2 - 3x - 2 using the long division method. The detailed steps show how to express the quotient and remainder for this advanced algebra problem. Ideal for Grades 10-12.

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