The solution hint given is to prove inductively, by noting that : $$ a^{n+1}b^{2n+2} = ab^2(a^nb^{2n}) = ab^2(a^nb^{3n} -1)$$
But, am unable to use the hint given, as am unable to see how $$ab^2(a^nb^{2n}) = ab^2(a^nb^{3n} -1)$$
I can only solve using the conventional inductive approach, as below:
Taking the base step, for $n=1$:$$ab^2 = ab^2$$
Taking the assumption that the hypothesis (to be proved) is true for the case $n=k$, get :$$a^kb^{2k} = (ab^2)^k,$$
Now, the final step to show that it is true for $n = k+1$:$$a^{k+1}b^{2k+2} = (ab^2)^{k+1} \,\,\,\,\,: (i)$$
For this take the l.h.s.: $a^{k+1}b^{2k+2} = a^k\cdot a\cdot b^{2k}\cdot b^2= (ab^2)^kab^2=$ desired r.h.s.