A strange equation I

Algebra Level 5

The four roots of the polynomial $$P(x)=x^{4}+8x+12$$ can be expressed (with the correct choice of signs) as:

$$x=\mp\sqrt{a\cos(d°)}\pm\sqrt{\pm\sqrt{b\sec(d°)}-c\cos(d°)}$$

$$0<d<40$$

Where, $$a$$, $$b$$, $$c$$ and $$d$$ are positive integers, and $$d$$ is in degrees. Find $$a+b+c+d$$.

You may also try Part II.

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