Is this 'Calculus' or 'Algebra'?

Calculus Level 5

I=0x5exdx=(2m4+m3+5m+9)!I=\int_0^{\infty} x^{5} e^{-x} dx = ( 2m^{4}+m^{3}+5m+9)!

Let the product of the real roots (of mm) of the equation above be PP .

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

(2a2b2c2)+(b+c)2a+1+(2b2a2c2)+(c+a)2b+1+(2c2b2a2)+(a+b)2c+1\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 3 decimal places\text{ 3 decimal places}

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

This question is part of the set Best of Me
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