Probability of Distinct Roots

Algebra Level 5

Let k k be a uniformly chosen real number from the interval (5,5) (-5, 5) . Let pp be the probability that the quartic f(x)=kx4+(k2+1)x2+k 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 1000p \lfloor 1000 p \rfloor ?

Details and assumptions

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

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