Factor. \( 6 a^{2}+12 a t+6 t^{2}-96 n^{2} \) \( 6 a^{2}+12 a t+6 t^{2}-96 n^{2}=\square \)

First, let's factor out the greatest common factor, which is 6, from the first three terms of the expression. That gives us: \( 6(a^2 + 2at + t^2) - 96n^2 \) or \( 6(a + t)^2 - 96n^2 \). Next, we can recognize this as a difference of squares: \( 6((a+t)^2 - 16n^2) \). This can be factored further into: \( 6((a+t) - 4n)((a+t) + 4n) \). So, the fully factored form is: \( 6((a+t) - 4n)((a+t) + 4n) \). Now, imagine applying this in real life! If you’re launching a small business (let's say, a bakery), understanding how to factor can help you break down your total costs into more manageable parts. Just like factoring a polynomial, you can identify fixed costs, variable costs, and profit margins, leading to clearer financial insights. For further study, exploring algebraic identities can be super beneficial! Dive into topics like the quadratic formula, completing the square, and polynomial long division. Websites such as Khan Academy and textbooks dedicated to algebra will provide you with problems and insights to sharpen your factoring skills!