Use polynomial long division to divide (3x^4 - 5x^3 + 2x^2 + 3x - 2) by (3x - 2).

To divide the polynomial \(3x^4 - 5x^3 + 2x^2 + 3x - 2\) by \(3x - 2\) using polynomial long division, follow these steps:

Step 1: Setup the division, writing the polynomial to be divided (dividend) inside the division symbol and the polynomial we’re dividing by (divisor) outside:

``` ___________________ 3x - 2 | 3x^4 - 5x^3 + 2x^2 + 3x - 2 ```

Step 2: Divide the first term of the dividend (\(3x^4\)) by the first term of the divisor (\(3x\)) to get the first term of the quotient. The result is \(x^3\).

Step 3: Multiply the divisor \(3x - 2\) by \(x^3\) and write the result below the corresponding terms of the dividend:

``` x^3 ___________________ 3x - 2 | 3x^4 - 5x^3 + 2x^2 + 3x - 2 - (3x^4 - 2x^3) ________________ - 3x^3 + 2x^2 ```

Step 4: Subtract the result from step 3 from the corresponding terms of the dividend:

``` x^3 ___________________ 3x - 2 | 3x^4 - 5x^3 + 2x^2 + 3x - 2 - (3x^4 - 2x^3) ________________ -3x^3 + 2x^2 ```

Step 5: Bring down the next term of the dividend to form a new sub-dividend:

``` x^3 ___________________ 3x - 2 | 3x^4 - 5x^3 + 2x^2 + 3x - 2 - (3x^4 - 2x^3) ________________ - 3x^3 + 2x^2 + 3x ```

Step 6: Repeat steps 2-5 for the new sub-dividend. Divide the first term of the new sub-dividend (\(-3x^3\)) by the first term of the divisor (\(3x\)) to get \(-x^2\), and continue the process:

``` x^3 - x^2 ___________________ 3x - 2 | 3x^4 - 5x^3 + 2x^2 + 3x - 2 - (3x^4 - 2x^3) ________________ - 3x^3 + 2x^2 + 3x + (3x^3 - 2x^2) ________________ 4x^2 + 3x ```

Step 7: Continue this process until you reach a sub-dividend that can no longer be divided by the divisor, which becomes your remainder:

``` x^3 - x^2 + 4x + 2 ___________________ 3x - 2 | 3x^4 - 5x^3 + 2x^2 + 3x - 2 - (3x^4 - 2x^3) ________________ - 3x^3 + 2x^2 + 3x + (3x^3 - 2x^2) ________________ 4x^2 + 3x - (4x^2 - 8/3x) ________________________ 17/3x - 2 ```

So the final answer is \(x^3 - x^2 + 4x + 2\) with a remainder of \(17/3x - 2\).