Let $f : R \to R$ be a continuous function such that for all real numbers $x$ and $y$, $$f(x+y)-f(x)-f(y) = xy+x^2y+xy^2$$and $f(6) = 69$. If $f(2\sqrt6) = a+b\sqrt{c}$, where $a$ and $b$ are integers and $c$ is a square-free integer, enter $a+b+c$.
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Tags: algebra, functional equation
Let $f : R \to R$ be a continuous function such that for all real numbers $x$ and $y$, $$f(x+y)-f(x)-f(y) = xy+x^2y+xy^2$$and $f(6) = 69$. If $f(2\sqrt6) = a+b\sqrt{c}$, where $a$ and $b$ are integers and $c$ is a square-free integer, enter $a+b+c$.