Given α and β are the roots of the quadratic equation px^2+qx+r=0, find the relationship between p,q and r if

a) α=2β+1

b) α=3β+1

Answers provided by the answer sheet are 2q^2 + pq - p^2 = 9pr and 3q^2 + 2pq - p^2 =16pr respectively.

As shown below, i just managed to solve a) but not b), in both solutions for b, i don't know what goes wrong and they are not same as the answer provided, it is possible to be more than one solution? Please help me