To divide the polynomial (3x^4 + 8x^3 + 8x - 3) by (3x - 1), we can use long division.

x^3 + 3x^2 + 4x + 3
___________________________

3x - 1 | 3x^4 + 8x^3 + 0x^2 + 8x - 3

(3x^4 - x^3)
9x^3 + 0x^2 + 8x
- (9x^3 - 3x^2)
___________________
3x^2 + 8x
- (3x^2 - x)
___________________
9x + 3
- (9x - 3)
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6

Therefore, (3x^4 + 8x^3 + 8x - 3) ÷ (3x - 1) = x^3 + 3x^2 + 4x + 3 with a remainder of 6.