is equal to :

Concept

For two quantities X and Y, X2 - Y2 factorises as (X + Y)(X - Y) - the difference-of-squares identity. This lets a difference of two squared expressions be rewritten as a product of their sum and their difference, which is often easier to simplify or cancel with other factors.

Application

1. Let X = (a - b + c) and Y = (a - b - c), so the numerator is X2 - Y2.

2. Sum: X + Y = (a - b + c) + (a - b - c) = 2(a - b).

3. Difference: X - Y = (a - b + c) - (a - b - c) = 2c.

4. By the identity, the numerator = (X + Y)(X - Y) = 2(a - b) x 2c = 4c(a - b).

5. The denominator is (b - a) = -(a - b), so the expression = 4c(a - b) / [-(a - b)] = -4c, valid whenever a is not equal to b.

Cross-check

Substituting a = 5, b = 2, c = 1: numerator = (5-2+1)2 - (5-2-1)2 = 42 - 22 = 16 - 4 = 12; denominator = (2-5) = -3; so the expression = 12 / (-3) = -4, which equals -4c for c = 1 - confirming the result independently.

Result

So the given expression simplifies to -4c.