If $ \frac{x^2 - 1}{x + 1} = 4 $, $ x = $

### Concept & Formula This problem tests your knowledge of basic algebraic factorization. Specifically, it uses the difference of two squares identity to simplify a rational expression. The core formula is: $$ a^2 - b^2 = (a - b)(a + b) $$ ### Step-by-Step Solution Examine the numerator of the given fraction: $ x^2 - 1 $. Since $ 1 = 1^2 $, this can be written as $ x^2 - 1^2 $. Apply the difference of squares formula to factor the numerator: $$ x^2 - 1 = (x - 1)(x + 1) $$ Substitute this back into the original equation: $$ \frac{(x - 1)(x + 1)}{x + 1} = 4 $$ Cancel the common term $ (x + 1) $ from the numerator and denominator: $$ x - 1 = 4 $$ Add 1 to both sides to solve for $ x $: $$ x = 4 + 1 $$ $$ x = 5 $$ ### Exam Strategy & Shortcut Whenever you see a quadratic term like $ x^2 - y^2 $ divided by a linear term like $ x + y $ or $ x - y $, immediately factor the quadratic. The test maker intentionally designed the problem so the denominator will cancel out. Bypassing the cross-multiplication step saves time and reduces the risk of errors. ### Common Pitfall A common mistake is to immediately cross-multiply, resulting in $ x^2 - 1 = 4(x + 1) \Rightarrow x^2 - 1 = 4x + 4 \Rightarrow x^2 - 4x - 5 = 0 $. While factoring this quadratic equation will eventually give you $ x = 5 $ and $ x = -1 $ (which is extraneous), it is significantly slower and more prone to calculation mistakes. ### Final Answer **Therefore, the correct answer is 5.**