The problem asks us to simplify the given expression: $a^3 + a^2 - (b^2 + 1)a + b^2 - 1$.

1. Problem Description

The problem asks us to simplify the given expression: $a^3 + a^2 - (b^2 + 1)a + b^2 - 1$.

2. Solution Steps

First, expand the expression:

$a^3 + a^2 - (b^2 + 1)a + b^2 - 1 = a^3 + a^2 - b^2a - a + b^2 - 1$.

Now, rearrange the terms:

$a^3 + a^2 - a - b^2a + b^2 - 1$.

We can group the terms with $a$:

$(a^3 + a^2 - a) + (b^2 - b^2a - 1)$.

From the first group, factor out an $a$:

$a(a^2 + a - 1) + b^2(1-a) - 1$.

Now we regroup the original expression and factor.

$a^3 + a^2 - (b^2 + 1)a + b^2 - 1 = a^3 - a + a^2 - b^2 a + b^2 - 1 = a(a^2-1) + a^2 - 1 - b^2a + b^2 = a(a-1)(a+1) + (a^2-1) - b^2(a-1) = a(a-1)(a+1) + (a-1)(a+1) - b^2(a-1) = (a-1)[a(a+1) + (a+1) - b^2] = (a-1)[a^2+a+a+1-b^2] = (a-1)[a^2 + 2a + 1 - b^2] = (a-1)[(a+1)^2 - b^2]$.

We can use the difference of squares formula: $x^2 - y^2 = (x+y)(x-y)$.

$(a-1)[(a+1)^2 - b^2] = (a-1)(a+1+b)(a+1-b)$.

3. Final Answer

$(a-1)(a+1+b)(a+1-b)$