$I=\int_0^{\infty} x^{5} e^{-x} dx = ( 2m^{4}+m^{3}+5m+9)!$

Let the product of the real roots (of $m$) of the equation above be $P$ .

Given that $a+b+c=P$, for $(a,b,c)\in R^{+}$. Find the $\text{ Maximum}$ value of:

$\dfrac{ (2a^{2}-b^{2}-c^{2})+(b+c)^2}{a+1} + \dfrac{ (2b^{2}-a^{2}-c^{2})+(c+a)^2}{b+1} + \dfrac{ (2c^{2}-b^{2}-a^{2})+(a+b)^2}{c+1}$

**Details and Assumptions**:

$\bullet$ Give your answer approximately up-to$\text{ 3 decimal places}$

$\bullet$ The roots of the equation are not necessarily distinct.