Product of Roots of a $4$th-degree Polynomial

Question

Problem:

Let $a$, $b$, $c$, and $d$ be distinct real numbers such that \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 know we can find a polynomial that $a$ is a root of, then do the same for $b, c,$ and $d,$ but how would I continue to do this problem?

Answer 1

Hint:

$a,b,c,d$ are the solutions of equation $$x=\sqrt{4\pm\sqrt{5\pm x}}$$

Repeatedly square both sides and you should get a $4$th-degree polynomial.

And the product $abcd$ would be the constant term of the polynomial.

(Why? Consider the equation $(x-a)(x-b)(x-c)(x-d)=0$.)