\[(x^2+x+2)^2-(a-3)(x^2+x+2)(x^2+x+1)+(a-4)(x^2+x+1)^2=0\]

If \(a_1,a_2,a_3,....,a_n\) are \(n\) integral values of \(a\) for which the above equation has at least one real root, then find the value of \[ \dfrac{\displaystyle \sum_{i=1}^n (a_i)^2}{\displaystyle \sum_{j=1}^n (a_j)} \times n\]

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