$\rm P(X)= X^4+2X^3+(2+2k)X^2+(1+2k)X+2k=X^4+2X^3+2X^2+X+2k(X^2+X+1)$ $\rm =X(X+1)(X^2+X+1)+2k(X^2+X+1)=(X^2+X+2k)(X^2+X+1)$. Since the roots of $\rm X^2+X+1$ are not real we have that the real roots of $\rm P(X)$ are the real roots of $\rm R(X)=X^2+X+2k$. Let $\rm R(r_1)=R(r_2)=0,r_1\in \mathbb{R}$.Since $\rm r_1+r_2=-1\in \mathbb{R}$ we have that $\rm r_2\in \mathbb{R}$. So $\rm r_1^2+r_2^2=(r_1+r_2)^2-2r_1r_2=(-1)^2-2(-2013)=1+4026=4027$.