4x³+2x²-3x+1/(2x+1)(x-3)

Answer 1

what about it? The numerator has no factors in common with the denominator.

(4x^3+2x^2-3x+1)/((2x+1)(x-3)) = 2x+6 + (33x+19)/((2x+1)(x-3))
or, as partial fractions,
2x+6 + (118/7)/(x-3) + (-5/7)/(2x+1)