If $a$, $b$, $c$, $d$, and $e$ are constants such that for every $x > 0$:
\[
\frac{5x^4 - 8x^3 + 2x^2 + 4x + 7}{(x + 2)^4} = a + \frac{b}{x + 2} + \frac{c}{(x + 2)^2} + \frac{d}{(x + 2)^3} + \frac{e}{(x + 2)^4} \, ,
\]
then what is the value of $a + b + c + d + e$?

Let's think step by step and output the final answer within \boxed{}.
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