Most experienced mathematicians will immediately identify $$x = 0$$ as a solution to this equation, and then do the same division (factoring) as your student in order to obtain the other solution at $$x=2$$. The only difference is that the experienced mathematician has already done due diligence with respect to the possibility of erasing a root/dividing by zero.

I am all for dividing both sides by $$6x$$ here, with the caveat that when dividing by an unknown quantity you are responsible for first checking whether that unknown quantity is (or might be) zero. Your student is most of the way there - all that's left is to push the conclusion C from information A and B which they already seem to have.

A) I don't know what $$x$$ is (or $$6x$$, as is the case here)
B) I'm 'not allowed' to divide by zero

C) I don't know whether or not I'm allowed to divide by $$x$$ (because it might be zero)

There are useful things to think about in this neighborhood as well. A similar situation where the variable-including-divisor being equal to zero represents the only solution, then we end up producing an absurdity. Eg, $$6x = 4x$$ $$\frac{6x}{x} = \frac{4x}{x}$$ $$6 = 4$$ If we trust our teacher, or the textbook, to have provided us with a logically sound equation in the first place, then the absurdity reached here tells us that we've broken some rules in our working with it. Since the only thing we did was to divide by $$x$$, we can guess that dividing by $$x$$ introduced the absurdity. Since we know that dividing both sides of an equation by a quantity is fair game for any value other zero, we now know that $$x=0$$, etc...

Most experienced mathematicians will immediately identify $$x = 0$$ as a solution to this equation, and then do the same division (factoring) as your student in order to obtain the other solution at $$x=2$$. The only difference is that the experienced mathematician has already done due diligence with respect to the possibility of erasing a root/dividing by zero.

I am all for dividing both sides by $$6x$$ here, with the caveat that when dividing by an unknown quantity you are responsible for first checking whether that unknown quantity is (or might be) zero. Your student is most of the way there - all that's left is to push the conclusion C from information A and B which they already seem to have.

A) I don't know what $$x$$ is (or $$6x$$, as is the case here)
B) I'm 'not allowed' to divide by zero

C) I don't know whether or not I'm allowed to divide by $$x$$ (because it might be zero)