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?