Tessellate S.T.E.M.S - Mathematics - School - Set 1 - Problem 2

Algebra Level 3

You are allowed to do two operations to a quadratic polynomial with integer coefficients. You can either replace f(x)f(x) by f(x+a)f(x+a) for any integer aa or you can switch the positions of the leading coefficient and the constant term (i.e. you can replace ax2+bx+cax^2+bx+c by cx2+bx+acx^2+bx+a). You start with the polynomial f=x2+x+1f=x^2+x+1. Using the given operations, which of the following polynomials can you arrive at starting from ff?

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

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