Would the GCD of x^2+x+c and (x-a)^2+(x-a)+c always be 1?
Hint: if the roots of the first one are p and q, what are the roots of the second one?
So it seems that two polynomials have nonzero GCD when there is at leaste one root shared between the two. So any two polynomials of the form a(x-t)^2+b(x-t)+c and ax^2+bx+c must have GCD 1 since they wouldn't have any roots shared between them.
What about x2 + 3x + 2 = 0 and (x + 1)2 + 3(x + 1) + 2 = 0?
ok, I wasn't thinkng clearly at the time.
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