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.