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

Let $x^2+6x+7=y$.
Thus, $y^2+6y+7=x$, which gives
$$x^2+6x+7-(y^2+6y+7)=y-x$$ or
$$(x-y)(x+y+7)=0.$$
Thus, $x=y,$ which does not give real solutions, or
$$y=-x-7,$$ which gives
$$x^2+6x+7=-x-7.$$
Can you end it now?
