To simplify the expression \(\frac{x^4 + 8x^3 + 16x^2 - x - 18}{x + 3}\), we can perform polynomial long division.

### Steps for Polynomial Long Division:

1. **Divide the leading term of the dividend by the leading term of the divisor**:  
   \[
   \frac{x^4}{x} = x^3
   \]
   This is the first term of the quotient.

2. **Multiply the divisor \(x + 3\) by \(x^3\)** and subtract it from the original polynomial:
   \[
   (x^4 + 8x^3 + 16x^2 - x - 18) - (x^3(x + 3)) = (x^4 + 3x^3)
   \]
   \[
   (x^4 + 8x^3 + 16x^2 - x - 18) - (x^4 + 3x^3) = 5x^3 + 16x^2 - x - 18
   \]

3. **Repeat the process**: Divide \(5x^3\) by \(x\):
   \[
   \frac{5x^3}{x} = 5x^2
   \]
   Multiply the divisor \(x + 3\) by \(5x^2\):
   \[
   5x^2(x + 3) = 5x^3 + 15x^2
   \]
   Subtract from the current remainder:
   \[
   (5x^3 + 16x^2 - x - 18) - (5x^3 + 15x^2) = x^2 - x - 18
   \]

4. **Repeat again**: Divide \(x^2\) by \(x\):
   \[
   \frac{x^2}{x} = x
   \]
   Multiply \(x + 3\) by \(x\):
   \[
   x(x + 3) = x^2 + 3x
   \]
   Subtract:
   \[
   (x^2 - x - 18) - (x^2 + 3x) = -4x - 18
   \]

5. **Final division**: Divide \(-4x\) by \(x\):
   \[
   \frac{-4x}{x} = -4
   \]
   Multiply:
   \[
   -4(x + 3) = -4x - 12
   \]
   Subtract:
   \[
   (-4x - 18) - (-4x - 12) = -6
   \]

Thus, the quotient is:
\[
x^3 + 5x^2 + x - 4
\]
and the remainder is \(-6\). Therefore, the result of the division is:
\[
\frac{x^4 + 8x^3 + 16x^2 - x - 18}{x + 3} = x^3 + 5x^2 + x - 4 + \frac{-6}{x+3}
\]

Would you like further clarification or details on any step?

### Relative Questions:
1. How do you perform synthetic division for polynomials?
2. Can you explain how to factor polynomials using long division?
3. What is the Remainder Theorem, and how does it apply here?
4. How can we use the Factor Theorem to check if \(x+3\) is a factor of the polynomial?
5. What are other methods to divide polynomials aside from long division?

**Tip**: When dividing polynomials, always make sure to align terms by degree to avoid mistakes during subtraction.

This step-by-step guide explains how to simplify the polynomial division of (x^4 + 8x^3 + 16x^2 - x - 18) by (x + 3) using polynomial long division. Includes detailed explanations of key concepts like the Remainder Theorem and the Factor Theorem. Suitable for high school students in Grades 10-12.

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