Let the squares be a2 and b2.
The difference is a2-b2=(a-b)(a+b).
Let a-b=x+2z and a+b=2x+z.
Add these two equations:
2a=3x+3z, a=(3/2)(x+z).
Subtract the first equation from the second:
2b=x-z, b=(x-z)/2.
So a2=(9/4)(x+z)2 and b2=¼(x-z)2.
(9/4)(x+z)2-¼(x-z)2=(x+2z)(2x+z).