The question reads "Solve the following inequality and graph the solution on a number line, ${x}^{2} < -6x + 12$ "
First, when doing this, I put it all on one side to try and factor it out, but I don't believe I could've factored it out properly, so I went to use the quadratic formula and got an indeterminate answer. So, now I'm stuck on how to go along with this question to answer it...
You're right that you should start by writing all the terms on one side. Doing this we get $$ x^2 + 6x - 12 < 0. $$ The function $f(x) = x^2 + 6x - 12$ is a parabola when you graph it, so what we are looking for is if the parabola ever dips below the $x$-axis. All values of $x$ where the parabola is below the axis is part of the solution.
We can find the zeros of $f$ with the quadratic formula like you suggested: $$ x = \frac{-6 \pm \sqrt{84}}{2} = -3 \pm \sqrt{21}, $$ so the parabola crosses the $x$-axis at those values of $x$.
Now remember we want the values of $x$ where $x^2 + 6x - 12 < 0$. Here it's useful to know what a parabola looks like. Since the first term is a positive $x^2$, the parabola will open upwards, the solution set we're looking for is exactly the interval between the two zeros we solved for. So our solution is $$ -3 - \sqrt{21} < x < -3 + \sqrt{21}. $$ If you plot the function on Desmos or a graphing calculator it may be easier to see.