Find the rational root 4x^3-9x^2-10x+3=0

Comment 1:  Try rational roots formed by that +3 and that +4:
For example:

Some of the POSSIBLE rational roots include plus or minus 4/3, 4/1, 2/3, 2/1, and so on.  Typically one would choose one of these POSSIBLE roots to start with and divide using synthetic division.  If there is only a zero remainder, that would be an indication that the chosen possible root actually is a rational root of 4x^3-9x^2-10x+3=0.

Let's try the possible rational root 3/4.  This was strictly a random choice.

       
3/4  /  4   -2    -10    3
                3     3/4
       
          4     1   -9 1/4            (not finished, but this info is enough to warrant
                                            rejecting 3/4 as a rational root; there is a non-
                                            zero remainder here.)

check out other possibilities, such as -1/4, -1/2, etc., until you find a divisor that leaves no remainder.