The expression provided is:

$a^2 - 1 + 2b - b^2$

To simplify or analyze this expression, let's break it down and see if it can be factored or simplified further.

1. The expression $a^2 - 1$ can be factored as a difference of squares: $a^2 - 1 = (a + 1)(a - 1)$

2. This leaves the expression as: $(a + 1)(a - 1) + 2b - b^2$

However, without additional information or specific instructions (such as whether we're looking for a factorization in terms of $a$ and $b$), this may be as simplified as it can get in general terms.

Would you like more details on factoring techniques or have any specific questions on how to proceed with this expression?

Here are some related questions to consider:

1. Can this expression be factored further in a different way?
2. What would happen if specific values were assigned to $a$ and $b$?
3. How would this expression change if we wanted it in terms of one variable, $a$ or $b$?
4. Could we complete the square for terms involving $b$ in this expression?
5. How would this expression behave if graphed with $a$ and $b$ as variables?

Tip: Always look for recognizable patterns like the difference of squares when simplifying expressions.