Simplify: `a^2b(a^3-a+1)-a b(a^4-2a^2+2a)-b(a^3-a^2-1)`

Explanation

To simplify the expression \( a^2b(a^3 - a + 1) - ab(a^4 - 2a^2 + 2a) - b(a^3 - a^2 - 1) \), we will follow these steps:

### Step 1: Distribute each term
We will distribute \( a^2b \), \( -ab \), and \( -b \) across their respective expressions.

1. \( a^2b(a^3 - a + 1) = a^2b \cdot a^3 - a^2b \cdot a + a^2b \cdot 1 = a^5b - a^3b + a^2b \)
2. \( -ab(a^4 - 2a^2 + 2a) = -ab \cdot a^4 + ab \cdot 2a^2 - ab \cdot 2a = -a^5b + 2a^3b - 2a^2b \)
3. \( -b(a^3 - a^2 - 1) = -b \cdot a^3 + b \cdot a^2 + b = -a^3b + a^2b + b \)
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