"Solve the equation .... for $x$" simply means that you figure out for which value(s) of $x$ the equation holds true. So, for example, if I say:
"Solve the equation $2x=6$ for $x$", then you should say: "Ah, yes, that equation holds true for $x=3$". Why?
Because if you fill in $x=3$ you indeed get $2x = 2\cdot3 = 6$.
The equation would not hold for $x=4$, since for $x=4$ we get $2x = 2\cdot4 = 8 \not = 6$
(indeed, you can show that $x=3$ is the only value for $x$ that would make $2x=6$ true)
Also, notice that an equation can hold for multiple values of $x$. For example, take $x^2 = 1$. That equation holds true for $x = 1$, but also for $x = -1$. But it holds false for any other value of $x$.
Finally, notice that there need not be any $0$'s involved with these kinds of problems: there were no $0$'s in the two examples I just gave.
However, we often like to set things equal to $0$, because if you have something like what you have:
$$(x^2 + 6x -7)(2x^2 - 5x-3) = 0$$
then we can 'divide and conquer', since this equation will hold true if either
$$(x^2 + 6x -7) = 0$$ or
$$(2x^2 - 5x-3) = 0$$
and so now I just need to 'solve' those two simpler equations. In other words, setting one side of the equation to $0$ will often simplify matters. So, for example, I could have taken $2x=6$ and changed that into $2x-6=0$. And $x^2=1$ can be changed into $x^2-1=0$