Let's simplify this expression step by step.

A^2b × (-ab) × (-ab^2) × (-ab^2)

First, let's multiply the numbers:
(-1) × (-1) × (-1) × A^2 × a × a × b × b^2 × b^2

Multiplying the negatives:
1 × 1 × 1 × A^2 × a × a × b × b^2 × b^2

Simplify the constants:
1 × A^2 × a × a × b × b^2 × b^2

Multiply the variables with the same base:
A^2 × a^2 × b^5

The final simplified expression is A^2a^2b^5.