The vertical asymptote requires a factor of (x-4) in the denominator.
The double zero requires a factor of (x-1)² in the numerator.
At x=0, these factors will give a value of (-1)²/(-4) = -1/4, so another constant factor of -12 is needed to make the y-intercept be 3.
Your rational function is ...
... y = -12(x -1)²/(x -4)
