# Is this 'Calculus' or 'Algebra'?

**Calculus**Level 5

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

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.