$$ x^3+bx^2+cx+d =0$$

In the equation above, \(b, c,\) and \(d\) are constants. If the equation has roots \(-2,0, \text{ and } 4\), what is the value of \(b+c+d\) ?

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Given the roots, we can write the equation of the polynomial in factored form:

$$ a(x+2)(x-0)(x-4)=0 $$

Expanding:

$$ ax(x+2)(x-4)=0 $$ $$ ax(x^2-2x-8)=0 $$ $$ a(x^3-2x^2-8x)=0 $$

\(a\) must equal zero for this equation to match the given equation. Comparing our equation with the given:

$$ x^3+bx^2+cx+d = x^3-2x^2-8x $$ $$ b=-2, c=-8, d=0 $$ $$ b+c+d = -2+(-8)+0$$ $$ = \boxed{-10} $$