What is the mathematical meaning of this question?

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.

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.

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.