Since A,b and C are the roots of the eq, they satisfie they eq.
therefore, A^3=-qA-r
similarly, B^3=-qB-r
similarly, C^3=-qC-r
A^5=-qA^3-rA^2
B^5=-qB^3-rB^2
C^5=-qC^3-rC^2
therfore, A^5+b^5+c^5=-q(A^3+B^3=C^3)-r(A^2+B^2+C^2)
then you keep subbing in until you get it completely in terms of r...