Uniquely Awesome

\[\large{{\color{blue}n}{\color{red}+}{\color{blue}n}^{\color{green}2}{\color{red}+}{\color{blue}n}^{\color{green}3}{ \color{red}+}{\color{blue}n}^{\color{green}4} = {\color{orange}(}{\color{blue}n}{\color{red}+}{\color{red}2}{\color{orange})}{\color{brown}!}}\]

What integer value of \(n\) satisfies the equation above?

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