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Let's solve the division problem:

The given expression to divide is:

\[
\frac{11x^2 - 6 + 10x - 2x^4}{-2x^2 + 1}
\]

### Step 1: Arrange the Polynomial in Standard Form
Reorder the terms of the numerator \( 11x^2 - 6 + 10x - 2x^4 \) to match descending powers of \( x \):

\[
-2x^4 + 11x^2 + 10x - 6
\]

### Step 2: Perform Polynomial Long Division
We will divide the leading term of the numerator by the leading term of the denominator:

1. **Divide \( -2x^4 \) by \( -2x^2 \):**
\[
\frac{-2x^4}{-2x^2} = x^2
\]

2. Multiply \( x^2 \) by \( -2x^2 + 1 \):
\[
x^2 \cdot (-2x^2 + 1) = -2x^4 + x^2
\]

3. Subtract the result from the original polynomial:
\[
(-2x^4 + 11x^2 + 10x - 6) - (-2x^4 + x^2) = 10x^2 + 10x - 6
\]

4. **Divide \( 10x^2 \) by \( -2x^2 \):**
\[
\frac{10x^2}{-2x^2} = -5
\]

5. Multiply \( -5 \) by \( -2x^2 + 1 \):
\[
-5 \cdot (-2x^2 + 1) = 10x^2 - 5
\]

6. Subtract the result from the current polynomial:
\[
(10x^2 + 10x - 6) - (10x^2 - 5) = 10x - 1
\]

### Step 3: Write the Quotient and Remainder
The quotient is:

\[
x^2 - 5
\]

The remainder is:

\[
10x - 1
\]

### Final Answer
\[
\frac{-2x^4 + 11x^2 + 10x - 6}{-2x^2 + 1} = x^2 - 5 + \frac{10x - 1}{-2x^2 + 1}
\]

Would you like further details or have any questions?  
Here are some related questions:

1. What is the process for performing polynomial division in general?
2. How do you determine the quotient and remainder in polynomial division?
3. What are some strategies to simplify polynomial division?
4. Can polynomial division be used to find factors of polynomials?
5. How can polynomial division be applied to solve algebraic equations?

**Tip:** Remember to always reorder polynomials in descending powers of \( x \) to make division easier!

Divide: (11x^2 - 6 + 10x - 2x^4) ÷ (-2x^2 + 1) and write your answer in the form Quotient + Remainder/(-2x^2 + 1).

Learn how to divide the polynomial (11x^2 - 6 + 10x - 2x^4) by (-2x^2 + 1) using polynomial long division. This detailed solution includes the quotient and remainder, explaining each step of the process. Suitable for high school students in Grades 9-12.

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