Quartic no more!

Algebra Level 3

The quartic equation (x+1)(x+2)(x+3)(x+4) =5 contains a pair of real and a pair of imaginary roots. If the real roots can be expressed as:

√(a/b+√c) -a/d

where a,b,c, and d are positive integers. Find the sum of a,b,c, and d.

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