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 2 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?

**Answer Choices**

A. 10

B. 13

C. 19

D. 20

E. 23