I have been trying to solve this one problem from the Duke Math Meet, which does not provide a solution:
Find all solutions of $(x^2+7x+6)^2 + 7(x^2+7x+6) + 6=x$
At first I tried to factorize the polynomial, but always had the right hand side x remain, which was inconvenient. I then tried using the fact that one can write the left hand side as a composition of functions, and equated that with the inverse of the quadratic plugged into the function, but that was a very nasty equation with square roots, still ending up with a quartic.
What other solution paths are viable for this problem, and is there a way to factor the quartic?