Solve by completing the square. \( \begin{array}{l}x^{2}+2 x-65=0 \\ \text { Choose the appropriate process for rearranging the equation, if needed, before completing the square }(x+k)^{2}=d \text {. } \\ \text { (Type an integer or an improper fraction.) } \\ \text { A. } x^{2}+2 x-65=0 \Rightarrow \frac{1}{2} x^{2}+x-\frac{65}{2}=0 \Rightarrow \frac{1}{2} x^{2}+x+\square=\frac{65}{2} \\ \text { B. } x^{2}+2 x-65=0 \Rightarrow x^{2}+2 x=65 \Rightarrow x^{2}+2 x+\square=65 \\ \text { C. } x^{2}+2 x-65=0 \Rightarrow x^{2}+2 x=65 \Rightarrow x^{2}+2 x+\square=65+\square \\ \text { D. } x^{2}+2 x-65=0 \Rightarrow \frac{1}{2} x^{2}+x-\frac{65}{2}=0 \Rightarrow \frac{1}{2} x^{2}+x+\square=\frac{65}{2}+\square\end{array} \)

To solve the equation \( x^{2}+2 x-65=0 \) by completing the square, you need to rearrange it first. The best option here is to move the constant term to the other side, resulting in \( x^{2}+2 x=65 \). Then, to complete the square, we add \( (1)^2 \) (which is 1) to both sides, transforming it to \( (x+1)^2 = 66 \). Thus, the correct process to complete the square would follow option B: \( x^{2}+2 x-65=0 \Rightarrow x^{2}+2 x=65 \Rightarrow x^{2}+2 x+\square=65 \) where \(\square\) will be \(1\). Now, here’s some context to make it more colorful and fun! The technique of completing the square dates way back to ancient civilizations, especially in the realms of Arabic mathematicians who were the rock stars of algebra! They saved their legendary math skills for handling far more than just equations - they tackled geometry, astronomy, and even architecture! So when you complete the square, you're not merely solving an equation, you're continuing a thousands-year-old mathematical journey. For a real-world application, completing the square is super handy in physics, especially when dealing with projectile motion. Let’s say you’re trying to calculate the maximum height a ball will reach when tossed. By expressing your height formula in vertex form through completing the square, you’ll easily find that peak height! It’s like unlocking secrets hidden in equations!