**Moderator's edit**:

\(a^2(k^2 + 1)x^2 - a(k+2)x^2 + x^4 - 10x^3 - (k^2 -k - 1)x^2 + 175x + 650 = 0 \)

If the sum of two of all the integral roots of the equation above is \(-5\), find the values of \(a \forall k \in \mathbb R \).

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