Prove that for all a,b in a ring R, $\displaystyle (-a)b=-ab=a(-b)$

The hint says use: $\displaystyle a+(-a)=0$

Since any integer has a negative, we know $\displaystyle a+(-a)=0$

Multiple by b:

$\displaystyle (a+(-a))b=0(b)\Rightarrow ab+(-a)b=0$

How can I show $\displaystyle (-a)b=-ab=a(-b)$ from the above?