Everybody loves to root for a nuisance - 2

Algebra Level 5

Let $a$ , $b$ , $c$ , $d$ be real numbers such that $b-d \geq 12.5$ and all zeroes $\beta_1, \beta_2, \beta_3,$ and $\beta_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(\beta_1^2+1)(\beta_2^2+1)(\beta_3^2+1)(\beta_4^2+1)$ can take.

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