To perform the long division of \((6x^3 - 16x^2 + 17x - 6) \div (3x - 2)\), we'll follow a step-by-step process.

### Step 1: Set up the long division
Dividend: \(6x^3 - 16x^2 + 17x - 6\)  
Divisor: \(3x - 2\)

### Step 2: Divide the first term of the dividend by the first term of the divisor
\[
\frac{6x^3}{3x} = 2x^2
\]
This is the first term of the quotient.

### Step 3: Multiply the entire divisor by the first term of the quotient
\[
2x^2 \times (3x - 2) = 6x^3 - 4x^2
\]

### Step 4: Subtract the result from the original dividend
\[
(6x^3 - 16x^2 + 17x - 6) - (6x^3 - 4x^2) = -12x^2 + 17x - 6
\]

### Step 5: Repeat the process with the new dividend
Divide the first term of the new dividend by the first term of the divisor:
\[
\frac{-12x^2}{3x} = -4x
\]
This is the second term of the quotient.

### Step 6: Multiply the entire divisor by the new term of the quotient
\[
-4x \times (3x - 2) = -12x^2 + 8x
\]

### Step 7: Subtract again
\[
(-12x^2 + 17x - 6) - (-12x^2 + 8x) = 9x - 6
\]

### Step 8: Repeat the process with the new dividend
Divide the first term of the new dividend by the first term of the divisor:
\[
\frac{9x}{3x} = 3
\]
This is the third term of the quotient.

### Step 9: Multiply the entire divisor by the new term of the quotient
\[
3 \times (3x - 2) = 9x - 6
\]

### Step 10: Subtract again
\[
(9x - 6) - (9x - 6) = 0
\]
Since the remainder is zero, the division is exact.

### Final Answer:
\[
\text{Quotient: } 2x^2 - 4x + 3
\]
Thus, \((6x^3 - 16x^2 + 17x - 6) \div (3x - 2) = 2x^2 - 4x + 3\).

Would you like to go over more examples, or do you have any other questions?

### Related Questions:
1. How do you handle cases where the remainder isn't zero in polynomial long division?
2. Can you factor the divisor and dividend to simplify division?
3. What happens if the divisor is of a higher degree than the dividend?
4. How can you apply synthetic division to this problem?
5. What is the significance of the remainder in polynomial division?

### Tip:
When performing long division, always double-check your multiplication and subtraction steps to avoid errors, especially when dealing with polynomials of higher degrees.

Learn how to perform polynomial division for the expression (6x^3 - 16x^2 + 17x - 6) divided by (3x - 2) with a detailed step-by-step solution. This problem covers polynomial long division and its application in algebra. Suitable for Grades 10-12 students.

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