You are allowed to do two operations to a quadratic polynomial with integer coefficients. You can either replace $f(x)$ by $f(x+a)$ for any integer $a$ or you can switch the positions of the leading coefficient and the constant term (i.e. you can replace $ax^2+bx+c$ by $cx^2+bx+a$). You start with the polynomial $f=x^2+x+1$. Using the given operations, which of the following polynomials can you arrive at starting from $f$?

This problem is a part of Tessellate S.T.E.M.S.

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