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📰 ight) + 5 $. Simplify: $ g(u) = (u - 3)^2 + 6(u - 3) + 5 = u^2 - 6u + 9 + 6u - 18 + 5 = u^2 - 4 $. Thus, $ g(x^2 - 1) = (x^2 - 1)^2 - 4 = x^4 - 2x^2 + 1 - 4 = x^4 - 2x^2 - 3 $. Final answer: $oxed{x^4 - 2x^2 - 3}$ 📰 Question: For all real numbers $ x $ and $ y $, find the number of functions $ f : \mathbb{R} o \mathbb{R} $ such that $ f(x + y) = f(x) + f(y) + 2xy $. 📰 Solution: Assume $ f $ is quadratic: $ f(x) = ax^2 + bx + c $. Substitute into the equation: $ a(x + y)^2 + b(x + y) + c = ax^2 + bx + c + ay^2 + by + c + 2xy $. Expand and compare coefficients: $ ax^2 + 2axy + ay^2 + bx + by + c = ax^2 + ay^2 + bx + by + 2c + 2xy $. Matching terms: $ 2a = 2 \Rightarrow a = 1 $, and $ 2c = c \Rightarrow c = 0 $. Thus, $ f(x) = x^2 + bx $. Any real $ b $ satisfies the equation, so there are infinitely many solutions. Final answer: $oxed{\infty}$ 📰 This If Statement Trick Will Change How You Think Forever 425319 📰 Hero Fiennes Tiffin Movies 9890376 📰 Unhide Apps Manually This Simple Trick Will Save Your Privacy 1430771 📰 Sherlock Season 4 4947517 📰 5 Pikmin 2 Mastery The Finale That Changed The Entire Franchise Forever 2923239 📰 Suhas Subramanyam 5936960 📰 Microsoft Windows 8 Pro Surface Tablet The Ultimate Blend Of Productivity Power 8585984 📰 Unmarshaller Revealed The Shocking Power Of Decoding Hidden Truths 5364544 📰 Gbp Usd Forecast 9630554 📰 This Secret Lottery Prediction Just Beat Every Oddscan You Predict The Big Win 8631930 📰 Furaf The Hidden Trick Every Fan Is Talking About Yes You Need It 4192860 📰 Youtube Video Downloader 4K 898405 📰 Apt In St Louis Park Mn 9571441 📰 Rain Gif That Looks Like Its Pouring Liquid Sunshine Total Trabuzz 6617233 📰 Personal Responsibility And Work Opportunity Reconciliation Act 1088005