Let's simplify the expression step by step:

(a - a^2 + 2)/(a - 1) * (1 - 2a + a^2)/(a + 2)

Simplify each fraction separately:

(a - a^2 + 2)/(a - 1) = (a^2 - a + 2)/(a - 1) = [(a - 2)(a - 1)]/(a - 1) = a - 2

(1 - 2a + a^2)/(a + 2) = (a^2 - 2a + 1)/(a + 2) = [(a - 1)(a - 1)]/(a + 2) = a - 1

Simplify the overall expression:

(a - 2) * (a - 1) = a^2 - a - 2

Therefore, the simplified expression is: a^2 - a - 2.