Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $(x+1{)}^2$, the remainder is $x+3$, and when $P(x)$ is divided by the polynomial $x^2+1$, the remainder is $x+1$. 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?