Before you understand A Square  B Square : a^{2}  b^{2} = (a + b) (a – b), you are advised to read:
How to Multiply Variables ?
How to Multiply Polynomials ?
What is Exponential Form ?
How this identity of a^{2}  b^{2} = (a + b) (a – b) is obtained:
Taking RHS of the identity:
(a + b) (a – b)
Multiply as we do multiplication of two binomials and we get:
= a(a  b) + b(a  b)
= a^{2}  ab + ab  b^{2}
Solve like terms and we get:
= a^{2}  b^{2}
Hence, in this way we obtain the identity i.e. a^{2}  b^{2} = (a + b) (a – b)
Following are few applications of this identity.
Example 1: Solve 3a^{2}  2b^{2}
Solution: This proceeds as:
Given polynomial 3a^{2}  2b^{2} represents identity third i.e. a^{2}  b^{2}
Where a = 3a and b = 2b
Now apply values of a and b on the identity i.e. a^{2}  b^{2} = (a + b) (a  b) and we get:
3a^{2}  2b^{2} = (3a + 2b) (3a  2b)
Hence, 3a^{2}  2b^{2} = (3a + 2b) (3a  2b)
Example 2: Solve (6m + 9n) (6m – 9n)
Solution: This proceeds as:
Given polynomial (6m + 9n) (6m – 9n) represents identity third i.e. a^{2}  b^{2}
Where a = 6m and b = 9n
Now apply values of a and b on the identity i.e. a^{2}  b^{2} = (a + b) (a  b) and we get:
(6m + 9n) (6m – 9n) = (6m)^{2}  (9n)^{2}
Expand the exponential forms on the LHS and we get:
= 36m^{2}  81n^{2}
Hence, (6m + 9n) (6m – 9n) = 36m^{2}  81n^{2}

