<s>[INST] Evaluate $a+b+c+d$ If $a$, $b$, $c$, and $d$ are distinct integers such that
$$(x-a)(x-b)(x-c)(x-d)=4$$
has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$?  

I tried to analyze graphically by shifting the graph of
  $f(x)=(x-a)(x-b)(x-c)(x-d)$ four units downward but couldn't infer anything due to the random nature of $a$, $b$, $c$ and $d$.

 [/INST] The product of the four distinct integers $r-a$, $r-b$, $r-c$, and $r-d$ is $4$. Thus $r-a,r-b,r-c,r-d$ are, in some order, $-1$, $1$, $-2$, and $2$. 
We may assume it is in the order we gave. So $r-a=-1$, $r-b=1$, $r-c=-2$, $r+d=2$. Solve for $a,b,c,d$ and add. 
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