<s>[INST] Quadratic Functional equations. Suppose $f$ is a quadratic ploynomial, with leading cofficient $1$, such that $$f(f(x) +x) = f(x)(x^2+786x+439)$$ For all real number $x$. What is the value of $f(3)$?
 [/INST] Suppose $f(x) = x^2 + bx + c$
Then 
$$f(f(x)+x) = (f(x)+x)^2 + b(f(x)+x) + c = f(x)^2 + 2xf(x) + x^2 + bf(x) + bx + c = f(x)^2 + 2xf(x) + bf(x) + f(x) = f(x)(f(x) + 2x + b+ 1)$$
Equating factors yields
$$f(x) + 2x + b+ 1 = x^2 + 786 x + 439$$
Then letting $x=3$ we have
$f(3) = 2799-b$
Finding $b$ is a trivial matter by expanding out the previous equation.
There may be an indirect method for determining $b$ but this is as far as I can get without solving for $f(x)$.
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