<s>[INST] $x^4-(a+b+c+d)x^3-(a+b+c)x^2-(a+b)x-a=0$,then the number of integer roots in the equation is? 
Let a,b,c,d be natural numbers.Now consider an equation 
$x^4-(a+b+c+d)x^3-(a+b+c)x^2-(a+b)x-a=0$,then the number of integer
  roots in the equation is ?

Me: Plugged in $0,1,-1$.No use.At 0 the sign is negative and at infinity the sign is positive. At least one real root.Not sure if that will be integer though :/.Help please!
 [/INST] Suppose that the equation has an integer root $x$.  By @JimmyK4542's comment we have $x>0$; also, $x\mid a$ so $x\le a$.  Clearly
$$x-(a+b+c)<0$$
and so
$$d=\frac{x^4-(a+b+c)x^3-(a+b+c)x^2-(a+b)x-a}{x^3}<0\ .$$
This is a contradiction.
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