<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] Plugging in $x=4$ shows that $4$ can be written as product of four distinct integers. The only divisors of $4$ are $\pm1, \pm2,\pm4$.
The only way to obtain $4$ is as product $(-2)\cdot(-1)\cdot 1\cdot 2$. Then $(r-a)+(r-b)+(r-c)+(r-d)=-2+(-1)+1+2=0$, i.e. $a+b+c+d=4r$.
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