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📰 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: $oxed{\infty}$ 📰 Question: Find the center of the hyperbola $ 9x^2 - 18x - 16y^2 - 64y = 144 $. 📰 Deadwater Fell Cast 3685620 📰 What Battle La Didnt Want You To Knowincredible Details Youre Missing 8292475 📰 When Do The Next Fortnite Season Start 3234433 📰 Secrets Swapped In Steel The Sword Mysteries You Never Learned 2527986 📰 You Wont Believe This Hidden Feature In Vlc Movie Player For Mac Thatll Change Your Watching Game 2682876 📰 Matt V Stock Astound Userstop 5 Reasons To Invest Now Before Its Gone 6590836 📰 Hhs Portal Login Hack Access Your Healthcare Account Fast Securely 6195941 📰 Trucking Giant Teeters On Survival As Court Files Chapter 11 331707 📰 El Cerrito Bank Of America 6376869 📰 Breed Laboratory 6145637 📰 Best Robotic Pool Cleaner 2025 1293087 📰 Internet Explorer Browser Download 7745464 📰 How To Rboot Phone 5392760 📰 Camphor Oil 3740032 📰 Ishares Bitcoin Etf 5244091