Let a polynomial with integer coefficients be nice if

- this polynomial has integer roots;
- its derivative has also integer roots.

For instance

$$p(x)=x(x-9)(x-24),\

p'(x)=3(x-4)(x-18)$$

is the smallest known nice cubic polynomial. Smallest here means a polynomial with the smallest absolute value of the largest coefficient ($9times24=216$). But how to verify with the help of MA that there are no smaller ones?