Everybody loves to root for a nuisance - 2

Algebra Level 5

Let aa, bb, cc, dd be real numbers such that bd12.5b-d \geq 12.5 and all zeroes β1,β2,β3,\beta_1, \beta_2, \beta_3, and β4\beta_4 of the polynomial P(x)=x4+ax3+bx2+cx+dP(x)=x^4+ax^3+bx^2+cx+d are real. Find the smallest value the product (β12+1)(β22+1)(β32+1)(β42+1)(\beta_1^2+1)(\beta_2^2+1)(\beta_3^2+1)(\beta_4^2+1) can take.

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