Let's stop at this point in your work:

\(\displaystyle (x-6)(x+3)>0\)

We know the original expression, the quadratic, has a graph that's a parabola opening upwards, so the positive portion of the graph will be to either side of the two roots, thus the solution set is:

\(\displaystyle (-\infty,-3)\,\cup\,(6,\infty)\)