Well done, Brian Smith!

Here's a program (unnecessary since it's been proved) which shows the same result for (a,b,c) = (-2,4,16)

def f(x,a,b,c):
    return x**3 + a*x**2 + b*x + c

bigNum = 100
for a in range(-bigNum, bigNum+1):
    if a == 0:
        continue
    for b in range(-bigNum, bigNum+1):
        if b == a or b == 0:
            continue
        for c in range(-bigNum, bigNum+1):
            if c == b or c == a or c == 0:
                continue
            if f(a,a,b,c) != a**3 or f(b,a,b,c) != b**3:
                continue
            print(a,b,c)