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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