What is the mathematical meaning of this question? 
$a,b,c \in\mathbb{Z}$ and $x\in\mathbb{R}$, then the following expression is always true:
$$(x-a)(x-6)+3=(x+b)(x+c)$$
Find the sum of all possible values of $b$.
A) $-8$
B) $-12$
C) $-14$
D) $-24$
E) $-16$

I didn't understand what is the meaning of   "...is always true".
Even though I can't understand the question, I wrote these:
$$(x-a)(x-6)+3=(x+b)(x+c) \Rightarrow x=\frac{6a-bc+3}{6+a+b+c}$$
Here, $b$ can take an infinite number of values. Or do I miss something? For example, let  random values $a=100,b=50,c=3$ then $x=\frac {151}{53}$.
Is there a problem with the question?
 A: Two polynomials which are always equal over the reals are exactly the same.  In this case, since $x$ is allowed to vary, while $a,b,c$ are fixed, these are two polynomials in $x$.
For them to be equal, the coefficients of $x$ must also be equal.  Therefore,
\begin{align}
-a-6&=b+c\\
6a+3&=bc.
\end{align}
Now, you can solve for $a$ in the first equation and substitute into the second equation, giving
$$
6(-b-c-6)+3=bc.
$$
The problem then becomes, for which integers does this equation have a solution?
If you solve for $b$ here, you'll get a fraction in $c$, which you can study to figure out which integers for $c$ result in integers for $b$.
The problem with your solution for $x$ is that the denominator of your fraction is zero.
A: To answer the explicit question, "Is there a problem with the question?," the answer is Yes, it's worded in a weird, nonsensical way. (I think this is why Dr. Sonnhard Graubner left a comment asking for the question's source: was it reproduced verbatim, or did the OP paraphrase the problem?)  A better version would be something like this: 

Consider the set of triples
  $(a,b,c)\in\mathbb{Z^3}$ for which the equation 
$$(x-a)(x-6)+3=(x+b)(x+c)$$ 
holds for all $x\in\mathbb{R}$. Find the sum of all the $b$'s among
  these triples.

A: The question is poorly worded. It should read something like this:

$a,b,c$ are integers such that the following equation holds for all
  $x\in\Bbb R$:

etc.
