To simplify the expression \(\frac{5x^3 - 7x^2 + 3x - 18}{x - 2}\), you can use polynomial long division. Here's the process:

### Step 1: Divide the first term
- Divide the leading term of the numerator \(5x^3\) by the leading term of the denominator \(x\):
  \[
  \frac{5x^3}{x} = 5x^2
  \]

### Step 2: Multiply and subtract
- Multiply the entire denominator \(x - 2\) by the result from step 1 \(5x^2\):
  \[
  5x^2 \cdot (x - 2) = 5x^3 - 10x^2
  \]
- Subtract this from the original polynomial:
  \[
  (5x^3 - 7x^2 + 3x - 18) - (5x^3 - 10x^2) = 3x^2 + 3x - 18
  \]

### Step 3: Repeat the process
- Divide the leading term of the new polynomial \(3x^2\) by \(x\):
  \[
  \frac{3x^2}{x} = 3x
  \]
- Multiply the denominator \(x - 2\) by this result \(3x\):
  \[
  3x \cdot (x - 2) = 3x^2 - 6x
  \]
- Subtract:
  \[
  (3x^2 + 3x - 18) - (3x^2 - 6x) = 9x - 18
  \]

### Step 4: Final division
- Divide \(9x\) by \(x\):
  \[
  \frac{9x}{x} = 9
  \]
- Multiply the denominator \(x - 2\) by this result \(9\):
  \[
  9 \cdot (x - 2) = 9x - 18
  \]
- Subtract:
  \[
  (9x - 18) - (9x - 18) = 0
  \]

Since the remainder is 0, the simplified expression is:

\[
\boxed{5x^2 + 3x + 9}
\]

Would you like more details on polynomial long division, or do you have any other questions?

Here are 5 related questions to expand on this topic:
1. How do you perform synthetic division for a polynomial?
2. What is the remainder theorem, and how does it relate to polynomial division?
3. Can you use polynomial division to factor polynomials completely?
4. What is the difference between polynomial long division and synthetic division?
5. How do you solve polynomial equations using division?

**Tip:** When dividing polynomials, always align similar terms (e.g., \(x^2\) with \(x^2\)) to avoid mistakes.

Learn how to simplify the polynomial (5x^3 - 7x^2 + 3x - 18)/(x - 2) using polynomial long division. This problem involves dividing polynomials and understanding quotient and remainder concepts. Suitable for high school students studying algebra.

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