Here's my method.Let the given polynomial be denoted as

f(x)=x⁶+ax³+bx²+cx+d then

f(-x)=x⁶-ax³+bx²-cx+d

Now,according to descartes rule of change of signs,the max. no. of +ve real roots are no. of sign changes in f(x) and max.no of -ve real roots are no. of sign changes in f(-x).

Now,If a,b,c,d>0 then no sign changes in f(x) and 4 in f(-x).So,A max. of 4 real roots can exist.

If a,b,c>0 and d<0 then 1 in f(x) and 3 in f(-x).So,again a max. of 4 real roots.

Similarly,you can observe that whatever may be the signs of a,b,c,d,You get a max. of 4 real roots to the eqn.So,All the roots of the eqn. can't be real.