The expression $(x^3+6x^2)-(5x^2+12x)$ can be rewritten as $(x+a)(x+b)(x+c)$, where $a,b,$ and $c$ are constants. What is the value of $a+b+c$?

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We want to expand and convert the polynomial into its factored form.

$$(x^3+6x^2)-(5x^2+12x)$$ $$=x^3+6x^2-5x^2-12x$$ $$=x^3+x^2-12x$$ $$=x(x^2+x-12)$$ $$=x(x+4)(x-3)$$

Comparing this expression to $(x+a)(x+b)(x+c)$,

$$a=0, b=4, c=-3$$ $$a+b+c=$$ $$0+4-3=\boxed{1}$$