A strange equation IV

Algebra Level 5

The five real roots of the polynomial \(P(x)=x^{5}-15x^{4}+70x^{3}-90x^{2}-55x-32\sqrt{2}+57\) can be expressed as: \(x_{1}=a-b\cos(d)\) , \(x_{2}=a-c\sqrt{c}\) , \(x_{3}=a+b\cos(e)\) , \(x_{4}=a+b\cos(f)\) , \(x_{5}=a+b\cos(g)\), \(x_{1}<x_{2}<x_{3}<x_{4}<x_{5}\) and \(0°<d,e,f,g<90°\). All the variables are positive integer numbers and the angles are in degrees. Find \(d+e+f+g-a-b-c\).

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