a) \( 3 p^{2} q+15 p q^{2} \)

Let's factor the expression \( 3 p^{2} q + 15 p q^{2} \). First, we can see that both terms share common factors. We can factor out the greatest common factor, which is \( 3 p q \). When we factor \( 3 p q \) out, we get: \[ 3 p q (p + 5 q) \] Now, we have simplified the expression into a product of \( 3 p q \) and \( (p + 5 q) \). This factored form provides a clearer look at the relationship between the terms. Next, did you know factoring is not just useful in algebra? It's a powerful tool in simplifying complex problems across various fields, like computer science, where efficient algorithms often rely on breaking problems down into smaller, manageable pieces. By identifying common patterns, you can often jump to solutions faster than trying every possible method!