Let's simplify the expression $ab(a - b) - a(a - b^2)$.

Step 1: Expand both parts of the expression

First, expand the term $ab(a - b)$:

$ab(a - b) = ab \cdot a - ab \cdot b = a^2b - ab^2$

Next, expand the term $a(a - b^2)$:

$a(a - b^2) = a \cdot a - a \cdot b^2 = a^2 - ab^2$

Step 2: Subtract the second expanded expression from the first

Now, substitute these expanded forms back into the original expression:

$a^2b - ab^2 - (a^2 - ab^2)$

Distribute the negative sign across the second expression:

$a^2b - ab^2 - a^2 + ab^2$

Step 3: Simplify the expression

Notice that the terms $-ab^2$ and $+ab^2$ cancel each other out:

$a^2b - a^2$

So, the simplified expression is:

$a^2b - a^2$

Would you like more details or have any questions?

Here are five related questions:

1. How would you factor the expression $a^2b - a^2$?
2. What is the expanded form of $a(b - c) - b(a - c)$?
3. How would you simplify the expression $ab(a - b) + ab(b - a)$?
4. What happens if you substitute specific values for $a$ and $b$ in the expression $ab(a - b) - a(a - b^2)$?
5. How would you approach simplifying a more complex expression like $ab(a - b) - a(a - b^2) + c(a - b^3)$?

Tip: Always double-check the distribution of negative signs when subtracting expressions to avoid common mistakes.