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📰 Solution: Assume $ f $ is quadratic: $ f(x) = ax^2 + bx + c $. Substitute into equation: $ a(x + y)^2 + b(x + y) + c = ax^2 + bx + c + ay^2 + by + c + 2xy $. Expand left: $ ax^2 + 2axy + ay^2 + bx + by + c $. Right: $ ax^2 + ay^2 + bx + by + 2c + 2xy $. Equate coefficients: $ 2a = 2 \Rightarrow a = 1 $, and $ 2c = c \Rightarrow c = 0 $. Thus, $ f(x) = x^2 + bx $. Check: $ (x + y)^2 + b(x + y) = x^2 + y^2 + 2xy + bx + by $, which matches. Any $ b $ works, so infinitely many solutions. \boxed{\infty}<think1. A train travels at a constant speed of 90 miles per hour. If it travels for 2.5 hours, how far does the train travel? 📰 The distance traveled by the train is given by the formula: 📰 Distance = Speed × Time 📰 How Microsoft Hidden Millions In Azure Openai Pricingare You Getting The Best Deal 898553 📰 5Portch Football Sim Hack Unleashes Legendary Gameplay No One Knows Exists 1015167 📰 Cast Of Power Book Iv Force 3315304 📰 How A Basket Brother Became The Hero Of My Lifeyoull Cant Trust This 3664492 📰 Hungarian Food 6605182 📰 Boost Your Gameplay Like Never Before The Sony Playstation 5 Pro Arrives Now 9941159 📰 April Birth Stone 6081332 📰 What Is The Best Bank To Bank With 459056 📰 Csulb Sso Breakdown Simple Login Associations Ultimate Time Saver 2418500 📰 American Horror Stories Season 13 6088045 📰 A B2 A B2 Quad Extif Real But Not Necessarily 9380303 📰 Wayne Residents Know This Fidelity Investments Drives Massive Growth Here 5569443 📰 Star Fox 64 The Hidden Features That Make This Nes Classic Unputdownable 5778987 📰 Alice Kessler 8496604 📰 People That Watch 8365073