<s>[INST] real solution of equation $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is 
Number of real solution of equation  $(x^2+6x+7)^2+6(x^2+6x+7)+7=x$ is 

Plan
Put $x^2+6x+7=f(x)$. Then i have $f(f(x))=x$
For $f(x)=x$
$x^2+5x+7=0$ no real value of $x$
For $f(x)=-x$
$x^2+8x+7=0$ 
$x=-7,x=-1$
Solution given is all real solution
Help me please
 [/INST] You are right that $f(f(x))=x$
This gives $f(x)=f^{-1}(x)$
Do you know anything about the relationship between a function and its inverse?
Try sketching both $y=f(x)$ and $y=f^{-1}(x)$ on the same axes...
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