I am trying to prove that $\pi+\pi^2$ is irrational assuming that $\pi$ is transcendental.
My work: I noticed that at least one of $\pi+\pi^2$ and $\pi-\pi^2$ is transcendental. (Because algebraic numbers are closed under addition.) However, similar methods probably cannot resolve the question, because otherwise they could be used to show that $\pi+e$ is irrational which is an open problem. This makes me think that a solution must use the fact that the numbers $\pi$ and $\pi^2$ are related.
If $\pi+\pi^2$ were rational then $\pi$ would be algebraic.