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The problem asks for the remainder of the division of the polynomial \( x^3 + 2x^2 - 2x + 6 \) by \( x^2 - 2x - 3 \).

To solve this, we use polynomial division:

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

2. **Step 2:** Now, divide the leading term of the new polynomial by the leading term of the divisor:
   \[
   \frac{4x^2}{x^2} = 4
   \]
   Multiply the entire divisor \( x^2 - 2x - 3 \) by \( 4 \):
   \[
   4(x^2 - 2x - 3) = 4x^2 - 8x - 12
   \]
   Subtract this from the polynomial:
   \[
   (4x^2 + x + 6) - (4x^2 - 8x - 12) = 9x + 18
   \]

The remainder is \( 9x + 18 \).

The correct answer is **option (c): 9x + 18**.

Would you like a detailed explanation or have any questions? Here are 5 related questions to explore further:

1. How do you find the quotient in polynomial division?
2. Can the remainder ever be of a higher degree than the divisor?
3. What is synthetic division and how does it compare to long division?
4. How do you determine if a polynomial is divisible by another polynomial?
5. What are some real-world applications of polynomial division?

**Tip:** Always check the degree of the remainder—it should be less than the degree of the divisor.

Sisa pembagian dari polinomial x^3 + 2x^2 - 2x + 6 dibagi x^2 - 2x - 3 adalah

Find the remainder when dividing the polynomial x^3 + 2x^2 - 2x + 6 by x^2 - 2x - 3 using polynomial long division. Suitable for Grades 10-12 students studying algebra and polynomial operations.

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