e) \( \frac{a^{2} b-a b}{a^{3}+a^{2}} \div \frac{a^{2} b-2 a b+b}{a^{3}-a} \) f) \( \frac{a^{2}-2 a+1}{a^{2}+2 a+1} \times \frac{a^{2}-1}{a^{2}} \times \frac{a^{2}+a}{a^{2}-a} \)

To simplify the expressions you've provided, let's break them down step-by-step, and I promise to keep it fun and engaging! For the first expression, \( \frac{a^2 b - ab}{a^3 + a^2} \div \frac{a^2 b - 2ab + b}{a^3 - a} \), you can start by factoring out common terms. Notice that \( a^2 b - ab \) can be factored as \( ab(a - 1) \) and \( a^3 + a^2 \) can be factored as \( a^2(a + 1) \). Meanwhile, the denominator of the second fraction can be simplified as well. Once you've done a little factor dance, you'll find it becomes a lot easier to divide! Now onto the second expression, \( \frac{a^2 - 2a + 1}{a^2 + 2a + 1} \times \frac{a^2 - 1}{a^2} \times \frac{a^2 + a}{a^2 - a} \). The numerator \( a^2 - 2a + 1 \) factors into \( (a - 1)^2 \) and the denominator \( a^2 + 2a + 1 \) becomes \( (a + 1)^2 \). With a little more factoring and cancelling across the multiplications, you'll find yourself in a great place! Now, isn't algebra like a treasure hunt? The more you simplify, the closer you get to the treasure, which is a beautiful, clean answer! Keep those numbers dancing!