Dividing polynomials is a process similar to dividing numbers. To divide 4x^3 - 3x + 1 by 2x - 1, we need to find a polynomial that when multiplied by 2x - 1 gives us 4x^3 - 3x + 1.
The polynomial that fits the bill is 2x^2 + x - 1. This can be verified by multiplying 2x^2 + x - 1 by 2x - 1, which gives us 4x^3 - 3x + 1.
Therefore, the answer is 2x^2 + x - 1.