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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 📰 Wild Hidden Object Challenges Thatll Blow Your Minddont Miss Out 2810638 📰 Global Trade Management Definition 1916016 📰 Ffie Stock Hype Is Realheres The Shocking Secret Behind Its Rising Power 6426772 📰 Bobbyism 5254502 📰 This Fidelity Investments Checking Account Will Change How You Manage Your Money Forever 4551873 📰 Aaronnesmith 5088868 📰 The Unbelievable Secret Behind Chole Bhature That Will Change How You Eat It Forever 2482605 📰 Windows 11 Desktop Widgets 8926419 📰 Destiny 2 30Th Anniversary Pack 6569514 📰 Dinosaurs Tv Show Ruins Everything We Thought We Known About Prehistoric Life 8704459 📰 Brisk Tea The Surprising Secret To Instant Energy Like Never Before 7206921 📰 The Untold Story Of Sofivishocking Twists That Shock Fans Forever 4953890 📰 Ipad At Verizon 6780977 📰 Killing Floor 3 Closed Beta 2156329 📰 Wells Fargo Bank Tenafly Nj 3835941