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📰 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 $. 📰 Solution: Group terms: $ 9(x^2 - 2x) - 16(y^2 + 4y) = 144 $. Complete the square: $ 9[(x - 1)^2 - 1] - 16[(y + 2)^2 - 4] = 144 $. Expand: $ 9(x - 1)^2 - 9 - 16(y + 2)^2 + 64 = 144 $. Simplify: $ 9(x - 1)^2 - 16(y + 2)^2 = 89 $. The center is at $ (1, -2) $. Final answer: $oxed 📰 Best Value Stocks 📰 Horror In The Heartland Michigan Shooting Shatters Peace Instantly 7597809 📰 Edge Surfing Game 📰 Hiset Test 4679226 📰 Saltillo Tile Magic Transform Your Home With This Cant Miss Look 630020 📰 Dont Miss These 2024 Films Before They Go Viral Time To Buy Tickets Now 1313044 📰 Download Gimp For Mac 8938211 📰 Stock Market News November 22 2025 📰 How Umb Online Banking Is Changing The Way You Manage Cash Forever 7358457 📰 Neat Tip Sncf Connect App Features Youll Use Every Commute 909700 📰 Android Read Receipts 📰 Youll Never Guess How Pajamas Trousers Blow Up Fashion In 2024 6912306 📰 Youve Got Smile Lines But Think Theyre Just From Agethink Again 2110605 📰 Neck Tattoos For Men 7 Eye Catching Designs You Actually Need In 2024 4491529 📰 Assassination By Firearm 9890349