Quartic with no rational solutions

Algebra Level 5

\[ x^4 - 2x^2 - 8x - 3 = 0 \]

Let \(a,b,c\) and \(d\) denote the roots of the equation above. And suppose we denote \(2\sqrt n + 2\sqrt m = |a| + |b| + |c| + |d| \) for positive integers \(m\) and \(n\), find the value of \(m^2 + n^2 \).

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