2022 AMC 10B Problem 21

Let P(x) be a polynomial with rational coefficients such that when P(x) is divided by the polynomial x^2 + x + 1, the remainder is x + 2, and when P(x) is divided by the polynomial x^2 + 1, the remainder is 2x + 1. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?

\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.