Problem - Rings (Herstien)

Q: $\displaystyle R$ is a ring with following property

1. It has a unit element, 1.

2. $\displaystyle (ab)^2 = a^2b^2 \forall a,b \in R $

Prove R is commutative.

(Source: Herstein - Problem#22 Pg 168 Ch 3 Ring Theory)

I did it in a very round about way. First proving bab = abb, aba=aab and then the final result ab = ba.

I essentially used

(a+1)(b+1)(a+1)(b+1) = (a+1)(a+1)(b+1)(b+1)

holds true for this ring.

Not too happy with my attempt. I think I just got the final result by a fluke. Is there any better/quick method / structured approach for this problem.