I need some help with the following exercise (number 15.10) of the book by Cox
First part: Let $O$ be an order in a quadratic field such that $O=[1,a\tau_1]$ where $\tau_1$ is a root of the polynomial $ax^2+bx+c$ with $a,b,c$ integers relatively prime and $a>0$. Prove that $a(\tau_1-1)O=[a(\tau_1-1),a(a+b+c)]$
I tried by hand but at certain point I don't know how to continue:$a(\tau_1-1)O=a(\tau_1-1)[1,a\tau_1]=[a(\tau_1-1),a(a\tau_1^2-a\tau_1)]=[a(\tau_1-1),a(-b\tau_1-c-a\tau_1)]=[a(\tau_1-1),a(a\tau_1+b\tau_1+c)]$
Can someone help me to conclude?
almost there: $$a(a\tau_1+b\tau_1+c)-(a+b)a(\tau_1-1)=a(a+b+c)$$