Let $f(x)=x^2+12x+30$. Solve $f(f(f(f(f(x)))))=0$ Here is my solve, is it correct?
I figured out that we can restate $f(f(x))$ as 
$((x+r)(x+s)+r)((x+r)(x+s)+s)$ 
thus
 $f(f(f(f(f(x)))))=0$ 
is 
$(x+r)^2(x+s)^2(4s+3r)(4r+3s)$
from vieta's
$0=(x+r)(x+s)(36^2-r^2)$ and we get $x=-6$
Did I make a mistake somewhere?
I'd also like to see other solutions
EDIT: How can I solve it using vieta then?

We have $f(x) = (x+6)^2 - 6$, therefore $f^2 (x) = (x+6)^4 - 6$, $f^3 (x) = (x+6)^8 - 6$, and so on.
So $f^5 (x) = (x+6)^{32} - 6 = 0$, giving the (real) solutions $x=-6 \pm \sqrt[32]{6}$.
