The factorization theorem says if there are rational roots, they would be the factors of 6, or you can try factors like (x+1), (x-1), (x-2), (x+2), (x-3),(x+3),(x-6),(x+6).
Since it can be shown that x+1 and x-1 are not factors, there are two out of the remaining six are zeroes.
Let f(x)=x^4 - 6x^2 - 7x - 6 , you can try
f(2)=2^4-6*2^2-7*2-6=16-24-14-6=-28
But since f(-2)=16-24+14-6=0, we conclude that x=-2 is a zero.
Continue trying f(3) [because -2*3=-6, the constant term to see if x=3 is a zero.