The potential rational solutions are the positive and negative fractions whose
numerator is a factor of the constant term, 30, and whose denominator is a
factor of the coefficient of the largest power of x, which is 1.

Potential rational solutions: 

Plot it on your TI-83 or 84:

Press WINDOW and make it read:

WINDOW
 Xmin=-10
 Xmax=10
 Xscl=1
 Ymin=-100
 Ymax=100
 Yscl=1

Press Y= and make it read:
\Y1=X^4+3X^3-13X^2-9X+30

Press GRAPH

You get something like this:



The rational solutions are the x-coordinates of the x-intercepts.

You see that there is an x-intercept at -5, so we use synthetic

division:

-5 | 1    3  -13   -9   30
   |     -5   10   15  -30
     1   -2   -3    6    0

This factors the left side as



It appears to have a root at 2:

 2 | 1   -2   -3    6   
   |      2    0   -6      
     1    0   -3    0

This further factors as



We set each factor on the left = 0:

x+5=0;    x-2=0;    x²-3=0
  x=-5      x=2;      x²=3
                       x=±√3

The four solutions are -5, 2, √3, -√3

Edwin