\( a^{2}+2 a q+q^{2}-16 \) \( a^{2}+2 a q+q^{2}-16=\square \) (Factor completely.)

To factor the expression \( a^{2}+2aq+q^{2}-16 \), observe that the first three terms form a perfect square trinomial. We can rewrite it as: \( (a + q)^2 - 16 \). Recognizing that \( 16 \) is \( 4^2 \), we can apply the difference of squares formula: \( (x^2 - y^2) = (x - y)(x + y) \). In our case, it's \( (a + q)^2 - 4^2 \): \[ ((a + q) - 4)((a + q) + 4) = (a + q - 4)(a + q + 4). \] Thus, the expression factors completely to: \[ (a + q - 4)(a + q + 4). \] Here's something fun: Did you know that the concept of factoring polynomials has roots that stretch back to ancient civilizations? Babylonian mathematicians were already working with quadratic equations over 3,000 years ago! They would use geometrical methods to solve problems that we now solve with algebra, showing that the love for numbers and shapes is timeless! For applying this knowledge, factoring is handy in various fields like engineering, physics, and economics. Understanding how to manipulate equations allows engineers to determine dimensions when designing structures, and economists can model financial trends or optimize profits. Factoring helps simplify problems, making them easier to solve and understand in the real world!