a) \( x^{4}-16 \)

Let's dive into factoring! The expression \( x^{4}-16 \) is a difference of squares, which can be factored neatly into \( (x^{2}+4)(x^{2}-4) \). The term \( x^{2}-4 \) can be factored further since it's also a difference of squares, giving us \( (x+2)(x-2) \). So, the complete factorization is \( (x^{2}+4)(x+2)(x-2) \). How fun is it to see the layers of factoring unfold? Now, let's spark some real-world application! Understanding how to factor polynomials like \( x^{4}-16 \) is foundational in various fields, such as engineering and physics. For instance, engineers often use polynomial equations to model structures and solve for critical points—a process vital for ensuring the safety and functionality of buildings and bridges. So the next time you tackle a polynomial, remember its practical importance!