step1

given 

x5-3x3+x2-3

step2

we can use factorization as under

x5-3x3+x2-3=  x^3(x^2-3) +1(x^2-3)

=  (x^3-3)(x^2-3)

step3

The two polynomials are irreducible . Therefore the splitting field must have subfields of degree 3 and of degree 2, so the degree of the splitting field over q is 5

roots are 

x^2=3

x= 3 or -3

and

x^3=3

here complex roots exists

3 ,3w ,3(1+w^2)

 hence total 5 roots 

two real and 3 complesx roots