Let's simplify each expression:

x^2 - 81
This is a difference of squares, so we can rewrite it as:
(x + 9)(x - 9)

y^2 - 4a + 4
This is a perfect square trinomial, so we can rewrite it as:
(y - 2)^2

36x^4y^2 - 169c^2
This is also a difference of squares, so we can rewrite it as:
(6x^2y + 13c)(6x^2y - 13c)

(x + 1)^2 - (x - 1)^2
This can be simplified using the difference of squares formula, which states that:
a^2 - b^2 = (a + b)(a - b)

Applying this formula:
[(x + 1) + (x - 1)][(x + 1) - (x - 1)]
= (2x)(2)
= 4x