Consider the polynomial below:

((x^2 - 85)^2 - 4176)^2 - 2880^2 = (x^2 - 1)*(x^2 - 11^2)*(x^2 - 13^2)*(x^2 - 7^2)

This polynomial could be considered a ''type 3'' polynomial,since it can be generated by only 3 squaring operations (ignoring the square on 2880 of course), and it has 2^3 distinct integer roots.

In general, if the integer ''2A'' can be written as a sum of two squares (ie x^2 + y^2) in at least two different ways, then there exists some ''C'' and ''D'' such that

((x^2 - A)^2 - B)^2 - C^2 = (x^2 - a^2)*(x^2 - b^2)*(x^2 - c^2)*(x^2 - d^2)

where ''a,b,c,d'' are all integers

Question:

Do there exist any ''type 4'' polynomials that have this property? That is a polynomial that can be generated in four squaring operations, that has 2^4 integer roots.