Find a quadratic polynomial P(x) with integer coefficients
that is not the square of a linear polynomial, such that
P(1), P(2), P(3), and P(4) are all perfect
squares.

Source: Joe Konhauser Problemfest

For such a polynomial, what is the largest n such that
P(1), P(2), P(3), ... and P(n) are
all perfect squares?