2x^3+13x^2-24x-16=0 has no rational roots, and so doesn't factorise. So how can we possibly find the roots? One way is to use systematic trial and error. You need a calculator for this. It can take a little time to get all three roots (a cubic equation has a maximum of three).
You can also draw a graph of the function y=2x^3+13x^2-24x-16 sufficiently accurately to see where the graph cuts (intercepts) the x axis. These are the roots. If you do this you may be able to see that the smallest root is between -8 and -7, another between -1 and 0, and another between 1 and 2.
Starting at x=-8, -1 or 1 calculate y and note the result. Then change the value by a small increment (e.g.,0.2) so you have x=-7.8, -0.8 or 1.2, and recalculate y. Note the result and particularly note where there is a sign change from positive to negative or vice versa. A sign change indicates that the root is between two values. For example, putting x=-1, y=19 and putting x=0, y=-16. So the sign change is positive to negative for the nearest root.
You can narrow the search once you know between what values a sign change occurs and then you can change the increment to the next decimal place. When I did this exercise for each estimated root I was able to find them to a higher accuracy: x=-7.8921, -0.5280 and 1.9200.