To solve the polynomial division \((3x^3 - 2x^2 + 4x - 5) \div (x - 2)\), we'll use long division.

### Step-by-Step Process:
1. **Divide the leading term** of the dividend (3x³) by the leading term of the divisor (x):
   \[
   \frac{3x^3}{x} = 3x^2
   \]
   Multiply \(3x^2\) by \((x - 2)\) and subtract it from the original polynomial.
   \[
   3x^3 - 2x^2 + 4x - 5 - (3x^3 - 6x^2) = 4x^2 + 4x - 5
   \]

2. **Divide the leading term** of the new polynomial (4x²) by the leading term of the divisor (x):
   \[
   \frac{4x^2}{x} = 4x
   \]
   Multiply \(4x\) by \((x - 2)\) and subtract:
   \[
   4x^2 + 4x - 5 - (4x^2 - 8x) = 12x - 5
   \]

3. **Divide the leading term** of the new polynomial (12x) by the leading term of the divisor (x):
   \[
   \frac{12x}{x} = 12
   \]
   Multiply \(12\) by \((x - 2)\) and subtract:
   \[
   12x - 5 - (12x - 24) = 19
   \]

### Final Result:
The quotient is \(3x^2 + 4x + 12\) with a remainder of 19. Hence:
\[
\frac{3x^3 - 2x^2 + 4x - 5}{x - 2} = 3x^2 + 4x + 12 + \frac{19}{x - 2}
\]

Would you like more details or further clarification?

Here are 5 related questions you might find interesting:
1. How do you perform synthetic division?
2. Can you divide polynomials when the divisor is not linear?
3. What are the applications of polynomial division in calculus?
4. How does the Remainder Theorem help in polynomial division?
5. Can you factor the cubic polynomial \(3x^3 - 2x^2 + 4x - 5\)?

**Tip:** Always arrange the polynomial terms in descending order before performing polynomial division!

Learn how to solve the polynomial division (3x³ - 2x² + 4x - 5) ÷ (x - 2) using long division. This detailed solution walks through the process of dividing a cubic polynomial by a linear divisor, providing insights into key algebraic techniques and the Remainder Theorem. Suitable for students in Grades 9-12.

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