Factor the trinomial completely. If this trinomial contains a greatest common factor (other than 1 ), don't forget to factor out the GCF first. \( x^{2}+2 x y-24 y^{2} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x^{2}+2 x y-24 y^{2}= \) B. \( x^{2}+2 x y-24 y^{2} \) is prime.

To factor the trinomial \( x^{2}+2xy-24y^{2} \), we'll start by searching for two numbers that multiply to \(-24\) (the product of \(a\) and \(c\) terms, where \(a=1\) and \(c=-24y^{2}\)) and add up to \(2\) (the coefficient of the \(xy\) term). The numbers \(6\) and \(-4\) fit the bill since \(6 \cdot (-4) = -24\) and \(6 + (-4) = 2\). Now we can rewrite the middle term: \[ x^{2} + 6xy - 4xy - 24y^{2} \] Next, we group the terms: \[ (x^{2} + 6xy) + (-4xy - 24y^{2}) \] Factoring out common factors from each group gives us: \[ x(x + 6y) - 4y(x + 6y) \] Now factor by grouping, leading to: \[ (x - 4y)(x + 6y) \] So, the complete factorization is: \[ x^{2}+2xy-24y^{2} = (x - 4y)(x + 6y) \] Choose option A: A. \( x^{2}+2xy-24y^{2} = (x - 4y)(x + 6y) \) --- Did you know that factoring trinomials like this has been a mathematical practice for centuries? Ancient mathematicians were already utilizing basic concepts of algebra to simplify complex problems, paving the way for future discoveries! In real-world applications, factoring expressions can be incredibly useful, especially in fields like physics and engineering. For instance, when you are calculating trajectories or optimizing surfaces, you often encounter quadratic equations where factoring can help break down complex relationships into manageable parts. Keep practicing, and you'll find it a valuable tool!