<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] Rearrange:
$$(x^2+6x+7)^2+6(x^2+6x+7)+7=x \iff \\
((x+3)^2-2)^2+6((x+3)^2-2)+10=x+3 \iff \\
(x+3)^4+2(x+3)^2-(x+3)+2=0 \iff \\
(x+3)^4+(x+3-\frac12)^2+(x+3)^2+\frac74=0.$$
The LHS is positive, hence, there is no real solution.
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