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The problem in the image asks for the quotient and remainder when dividing the polynomial \( 2x^4 - 2x^2 + 8x + 5 \) by \( x^2 + 2 \).

To solve this, we will use **polynomial long division**:

### Step 1: Set up the division

We are dividing:

\[
\frac{2x^4 - 2x^2 + 8x + 5}{x^2 + 2}
\]

### Step 2: Divide the leading terms

- The leading term of the dividend is \( 2x^4 \), and the leading term of the divisor is \( x^2 \).
- Divide \( 2x^4 \) by \( x^2 \), which gives \( 2x^2 \).

### Step 3: Multiply and subtract

Multiply \( 2x^2 \) by \( x^2 + 2 \), which gives:

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

Now subtract this from the original polynomial:

\[
(2x^4 - 2x^2 + 8x + 5) - (2x^4 + 4x^2) = -6x^2 + 8x + 5
\]

### Step 4: Divide the new leading term

- The new leading term is \( -6x^2 \).
- Divide \( -6x^2 \) by \( x^2 \), which gives \( -6 \).

### Step 5: Multiply and subtract

Multiply \( -6 \) by \( x^2 + 2 \), which gives:

\[
-6x^2 - 12
\]

Now subtract this from the current polynomial:

\[
(-6x^2 + 8x + 5) - (-6x^2 - 12) = 8x + 17
\]

### Step 6: Write the final quotient and remainder

The quotient is \( 2x^2 - 6 \) and the remainder is \( 8x + 17 \).

### Final Answer:

- **Quotient**: \( 2x^2 - 6 \)
- **Remainder**: \( 8x + 17 \)

Would you like a detailed explanation of polynomial long division, or do you have any other questions?

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Here are five related questions:
1. How do you perform synthetic division for polynomials?
2. What are the differences between synthetic and long division in polynomials?
3. How do you handle division when the remainder is zero?
4. Can polynomial division be used to factor polynomials?
5. How does division of polynomials relate to the Remainder Theorem?

**Tip**: Always align terms of the same degree while performing polynomial long division to avoid confusion.

For the given expression, find the quotient and the remainder. The expression is 2x^4 - 2x^2 + 8x + 5 divided by x^2 + 2.

Solve the polynomial division of 2x^4 - 2x^2 + 8x + 5 divided by x^2 + 2. This detailed solution explains the steps of polynomial long division, including calculating the quotient and remainder. Suitable for Grades 10-12.

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