Divide the polynomial \(3x^3 - 2x^2 + 5x - 8\) by \(x - 2\).

Step 1 ：Set up the long division: \(\frac{3x^3 - 2x^2 + 5x - 8}{x - 2}\)

Step 2 ：Divide the first term in the numerator by the first term in the denominator, \(\frac{3x^3}{x} = 3x^2\). Write this above the division line.

Step 3 ：Multiply \(3x^2\) by \((x - 2)\) to get \(3x^3 - 6x^2\). Write this beneath \(3x^3 - 2x^2\) and subtract to get \(4x^2\).

Step 4 ：Bring down the next term from the numerator to get \(4x^2 + 5x\).

Step 5 ：Divide \(4x^2\) by \(x\) to get \(4x\). Write this above the division line next to \(3x^2\).

Step 6 ：Multiply \(4x\) by \((x - 2)\) to get \(4x^2 - 8x\). Write this beneath \(4x^2 + 5x\) and subtract to get \(13x\).

Step 7 ：Bring down the next term from the numerator to get \(13x - 8\).

Step 8 ：Divide \(13x\) by \(x\) to get \(13\). Write this above the division line next to \(3x^2 + 4x\).

Step 9 ：Multiply \(13\) by \((x - 2)\) to get \(13x - 26\). Write this beneath \(13x - 8\) and subtract to get \(18\).

Step 10 ：Since there are no more terms to bring down from the numerator, this is the remainder. Write this as \(\frac{18}{x - 2}\) next to the quotient.

Step 11 ：So, \(3x^3 - 2x^2 + 5x - 8\) divided by \(x - 2\) is \(3x^2 + 4x + 13 + \frac{18}{x - 2}\).