F4 Nature of roots

Since r is a root of the quadratic equation x^2 - 3x - 5 = 0,
i.e. r^2 - 3r - 5 = 0
r^5-3r^4-5r^3+2r^2-6r
= r^3 (r^2 - 3r - 5) + 2r^2 - 6r - 10 + 10
= r^3 (r^2 - 3r - 5) + 2(r^2 - 3r - 5) + 10
= 0 + 0 + 10
= 10