Archit's Challenge 7

Algebra Level 5

For real numbers \(a,b,c,d,p,q,r,s\) with \(b-d\geq 5 \), let the roots of the quartic polynomial \(f(x) = x^4 + ax^3 + bx^2 + cx + d \) be \(p,q,r\) and \(s\). Find the minimum value of

\[ (1+p^2)(1+q^2)(1+r^2)(1+s^2). \]

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