Multiplying and Dividing Rational Expressions, in Depth

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

(Hint: Multiply through by $(x+2)^4$, and plug in a nice value of $x$. Why can we do this?)

Source: Math Prize for Girls