How do I factor 3X^3-19X^2-14X=0?

Just as you look at it, the only factor common to all three terms is 'x'.
So you can factor the 'x' out and make the equation . . .

           x (3x²-19x - 14)  =  0

Right there, that tells you that [ x = 0 ] is a solution of the original equation.

But you're not looking for solutions, you're looking for factors, so you continue.

What about that ugly trinomial:  [ 3x² - 19x - 14 ]  ?  Can that be factored ?

Not easily.  But if you stuff it into the quadratic equation, you'll find two solutions
to the equation
                                 3x² - 19x - 14 = 0

and you'll recall that  (x - a solution) is a factor of the left side.
So you can find the remaining two factors of the original trinomial.