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Solve the polynomial using normal long division 

The expression you've provided for division is:

\[
\frac{x^4 + 3x^3 + 0x^2 + 6x - 4}{x^2 + 2}
\]

We will solve this using the method of polynomial long division.

### Steps for Long Division:

1. **Divide the first term**:  
   \[
   \frac{x^4}{x^2} = x^2
   \]
   Multiply \(x^2\) by the divisor \(x^2 + 2\):
   \[
   x^2(x^2 + 2) = x^4 + 2x^2
   \]
   Subtract this result from the original polynomial:
   \[
   (x^4 + 3x^3 + 0x^2 + 6x - 4) - (x^4 + 2x^2) = 3x^3 - 2x^2 + 6x - 4
   \]

2. **Divide the next term**:  
   \[
   \frac{3x^3}{x^2} = 3x
   \]
   Multiply \(3x\) by \(x^2 + 2\):
   \[
   3x(x^2 + 2) = 3x^3 + 6x
   \]
   Subtract this result:
   \[
   (3x^3 - 2x^2 + 6x - 4) - (3x^3 + 6x) = -2x^2 - 4
   \]

3. **Divide the next term**:  
   \[
   \frac{-2x^2}{x^2} = -2
   \]
   Multiply \(-2\) by \(x^2 + 2\):
   \[
   -2(x^2 + 2) = -2x^2 - 4
   \]
   Subtract this result:
   \[
   (-2x^2 - 4) - (-2x^2 - 4) = 0
   \]

### Final Quotient:
\[
x^2 + 3x - 2
\]

Thus, the result of the division is:

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

Let me know if you would like further details or have any questions!

Here are 5 related questions:
1. How do you handle remainders in polynomial division?
2. Can polynomial division result in a remainder that isn't zero?
3. How does synthetic division differ from long division?
4. What is the connection between polynomial division and factorization?
5. How can you use polynomial long division to simplify rational expressions?

**Tip**: Always ensure that both the dividend and divisor are written in descending order of powers before starting polynomial division!

Solve the polynomial using normal long division.

Learn how to solve the polynomial \(\frac{x^4 + 3x^3 + 0x^2 + 6x - 4}{x^2 + 2}\) using the long division method. This step-by-step guide covers core algebraic concepts of polynomial division, suitable for students in Grades 10-12.

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