Probability of Distinct Roots

Algebra Level 5

Let \( k \) be a uniformly chosen real number from the interval \( (-5, 5) \). Let \(p\) be the probability that the quartic \( f(x) = kx^4 + (k^2+1)x^2 + k \) has 4 distinct real roots such that one of the roots is less than -4, and the other 3 roots are greater than -1. What is the value of \( \lfloor 1000 p \rfloor \)?

Details and assumptions

\( \lfloor x \rfloor \) refers to the greatest integer function, which gives the largest integer that is smaller than or equal to \(x\). For example, \( \lfloor 3 \rfloor =3 \), \( \lfloor \sqrt{2} \rfloor =1\), \( \lfloor -\pi \rfloor=-4 \).

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