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{\inf
["Distance Traveled by a Train Moving at Constant Speed", "If a train travels at a constant speed of 90 miles per hour for 2.5 hours, the distance covered can be calculated using the fundamental formula for constant speed:", "[\n\ ext{Distance} = \ ext{Speed} \ imes \ ext{Time}\n]", "Given:\n- Speed = 90 miles per hour\n- Time = 2.5 hours", "Substitute the values into the formula:", "[\n\ ext{Distance} = 90 \ imes 2.5\n]", "Calculate the product:", "[\n90 \ imes 2.5 = 225\n]", "Therefore, the train travels 225 miles in 2.5 hours.", "This simple application of the speed-distance-time relationship is widely used in everyday travel plans, logistics, and transportation engineering to estimate travel time or distance efficiently.", "\boxed{225} miles"]
