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To solve this division, we are dividing a polynomial by another polynomial. Specifically, we need to divide:

\[
\frac{18x^2 - 3 + 8x - 10x^4}{-5x^2 - 1}
\]

and express the result in the form:

\[
\text{Quotient} + \frac{\text{Remainder}}{-5x^2 - 1}.
\]

### Step-by-Step Solution

1. **Rewrite the Dividend in Standard Form**: Arrange terms of \(18x^2 - 3 + 8x - 10x^4\) in descending order of powers of \(x\), so it becomes:
   \[
   -10x^4 + 18x^2 + 8x - 3.
   \]

2. **Perform Polynomial Division**:
   - The leading term in the dividend is \(-10x^4\), and the leading term in the divisor is \(-5x^2\).
   - To get the first term in the quotient, divide \(-10x^4\) by \(-5x^2\), which gives \(2x^2\).
   - Multiply \(2x^2\) by \(-5x^2 - 1\) and subtract this result from the original polynomial.

3. **Continue Division**:
   - Subtract and repeat this process until you reach a remainder that has a degree less than the divisor's degree (\(-5x^2 - 1\)), which is 2.

After completing this process, the answer will take the form:

\[
\text{Quotient} + \frac{\text{Remainder}}{-5x^2 - 1}.
\]

Let me know if you would like the detailed steps or help with another problem. 

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### Follow-up Questions

1. How do you perform polynomial division in general?
2. Why is it important to arrange terms in descending powers during division?
3. What is the significance of the remainder in polynomial division?
4. Can polynomial division be used to simplify rational expressions?
5. How does polynomial division relate to synthetic division?

#### Tip
When performing polynomial division, always align terms by their degree for accuracy.

Divide (18x^2 - 3 + 8x - 10x^4) ÷ (-5x^2 - 1) and write the answer in the form: Quotient + Remainder / (-5x^2 - 1).

Learn how to divide the polynomial (18x^2 - 3 + 8x - 10x^4) by (-5x^2 - 1) and express the result in quotient and remainder form. This problem covers key concepts in polynomial division, ideal for students in Grades 10-12.

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