a) \( 15 a^{2} b-5 a b^{3}= \)

To factor the expression \( 15 a^{2} b - 5 a b^{3} \), we can start by identifying the greatest common factor (GCF) of the terms. The GCF of \( 15 a^{2} b \) and \( 5 a b^{3} \) is \( 5 a b \). Now we can factor out \( 5 a b \): \[ 15 a^{2} b - 5 a b^{3} = 5 a b (3 a - b^{2}). \] So, the final factored form of the expression is \( 5 a b (3 a - b^{2}) \). Want to know more about the beauty of algebraic factoring? It’s like solving a puzzle where you rearrange the pieces to make a clearer picture of the relationship between the terms! Plus, mastering these skills can really amp up your mathematical confidence. Additionally, did you know factoring is not just for algebra? It’s a pivotal technique in real-world applications such as engineering and economics, where breaking down complex problems into simpler, manageable parts is essential for effective analysis and solution development. It's all about finding the right pieces to make the big picture work better!