An algebra problem by Krishna Sharma

Algebra Level 5

Given that the sum of the maximum and minimum value of

$\frac{x^4}{x^8+2x^6-4x^4+8x^2+16}$

can be expressed in the form $\frac{p}{q}$ , where $p$ and $q$ are coprime, positive integers, find the value of $p+q$ .