Let supose that we want to find all polynomials H(x) such that :

H(nx) = H'(x)*H''(x) with n a positive integer!

If the rank of the H(x) = n then we find n = n-1 + n-2 and n=3

So H(x) = ax^3+bx^2+cx+d.

After calculus i find two solution :

1) n =1 then H(x) = 1/18*x^3 + a*x^2 + 6*a*x+12*a^3

were a is a real number.

2) n > 1 H(x) = n^3/18*x^3