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📰 Solution: Assume $f$ is quadratic. Let $f(x) = px^2 + qx + r$. Substitute into the equation: $p(a + b)^2 + q(a + b) + r = pa^2 + qa + r + pb^2 + qb + r + ab$. Expand and equate coefficients: $p(a^2 + 2ab + b^2) + q(a + b) + r = pa^2 + pb^2 + q(a + b) + 2r + ab$. Simplify: $2pab = ab + 2r$. For this to hold for all $a, b$, we require $2p = 1$ and $2r = 0$, so $p = rac{1}{2}$, $r = 0$. The linear term $q$ cancels out, so $f(x) = rac{1}{2}x^2 + qx$. Verifying, $f(a + b) = rac{1}{2}(a + b)^2 + q(a + b) = rac{1}{2}a^2 + ab + rac{1}{2}b^2 + q(a + b)$, and $f(a) + f(b) + ab = rac{1}{2}a^2 + qa + rac{1}{2}b^2 + qb + ab$. The results match. Thus, all solutions are $f(x) = oxed{\dfrac{1}{2}x^2 + cx}$ for some constant $c \in \mathbb{R}$.Question: A conservation educator observes that the population of a rare bird species increases by a periodic pattern modeled by $ P(n) = n^2 + 3n + 5 $, where $ n $ is the year modulo 10. What is the remainder when $ P(1) + P(2) + \dots + P(10) $ is divided by 7? 📰 Solution: We compute $ \sum_{n=1}^{10} P(n) = \sum_{n=1}^{10} (n^2 + 3n + 5) = \sum_{n=1}^{10} n^2 + 3\sum_{n=1}^{10} n + \sum_{n=1}^{10} 5 $. 📰 Using formulas: 📰 Georgia State Vs Vanderbilt 7847436 📰 Doc Hollywood Cast 6640363 📰 Hyatt Regency Savannah 6892103 📰 Naruto Storm Exposed The Hidden Technique No Fan Should Miss 7214150 📰 180C How Hot Is That The Shocking Fahrenheit Conversion You Need Now 520482 📰 How To Connect Skullcandy Headphones 8182924 📰 Ny Noodletown Chinatown 9766284 📰 Girly Pop Vs Everything Else The Catchy Sound Taking Over Tiktok 1110313 📰 Cbs Detroit 4304197 📰 These 7 Shocking Metric Windows Will Revolutionize Your Home Design 1302680 📰 Your Passwords Are Obsolete Heres Why Passwordless Access Is The Future 4463598 📰 Probability Of Failure Rolling 7 1 P 07 3464441 📰 Ive Gotten Strongerthis Transformation Will Give You Major Inspiration 8980579 📰 You Wont Believe Whats Inside This Movie Box For Your Pcshocked Optix 4857108 📰 Nuclear Explosion Range 551457