<s>[INST] Solving $x^5+x^4-12x^3-21x^2+x+5=0$ $$x^5+x^4-12x^3-21x^2+x+5=0$$ 
I think it can be solved by trigonometric ways, but how?
 [/INST] Inspecting the discriminant and the small primes where the polynomial splits suggests that its splitting field is the class field of modulus $31$ corresponding to the subgroup $\{\pm 1, \pm 5, \pm 6\}$ of $(\Bbb Z/31\Bbb Z)/\{\pm 1\}$.
Which means that the splitting field is $\Bbb Q\left(2\left(\cos\frac{2\pi}{31}+\cos\frac{10\pi}{31}+\cos\frac{12\pi}{31}\right)\right)$.
Then I computed the minimal polynomial of this quantity (pure curiosity), and ended up on your polynomial exactly.
So the roots are $2\left(\cos\frac{2\pi}{31}+\cos\frac{10\pi}{31}+\cos\frac{12\pi}{31}\right)$, and its conjugates $$2\left(\cos\frac{4\pi}{31}+\cos\frac{20\pi}{31}+\cos\frac{24\pi}{31}\right) , \\
2\left(\cos\frac{8\pi}{31}+\cos\frac{22\pi}{31}+\cos\frac{14\pi}{31}\right), \\
2\left(\cos\frac{16\pi}{31}+\cos\frac{18\pi}{31}+\cos\frac{28\pi}{31}\right),\\
2\left(\cos\frac{30\pi}{31}+\cos\frac{26\pi}{31}+\cos\frac{6\pi}{31}\right).$$
I have no idea on how to guess this without the heavy machinery.
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