<s>[INST] System of 4 quartic equations \begin{align*}a &=\sqrt{4+\sqrt{5+a}},\\ b &=\sqrt{4-\sqrt{5+b}},\\ c &=\sqrt{4+\sqrt{5-c}},\\ d &=\sqrt{4-\sqrt{5-d}}.\end{align*} 
Compute $abcd$.
I set up each as a quartic and got $a^4-8a^2-a+11=0$ and similar equations for $b,c,d$, but am unable to figure out how to format the roots as a product of all of them.  Could someone give me a bit of a hint?  Thank you.
 [/INST] We get that $a$, $b$, $-c$ and $-d$ are distinct roots of the polynomial
$$x^4-8x^2-x+11.$$
Hence,
$$x^4-8x^2-x+11=(x-a)(x-b)(x+c)(x+d)$$
so
$$abcd=11.$$
</s>