x4+15x3+70x2+120x+64=0 has coefficients 1, 15, 70, 120, 64.
If we add together the coefficients of even powers of x we get 1+70+64=135.
And, if we add together the remaining coefficients we get 15+120=135.
This shows that x=-1 is a root, which we can divide by, using synthetic division, to reduce the degree of the polynomial:
-1 | 1 15 70 120 64
1 -1 -14 -56 | -64
1 14 56 64 | 0 = x3+14x2+56x+64.
Note that x3+a3 can be factorised: (x+a)(x2-ax+a2), so if a=4:
x3+14x2+56x+64=x3+64+14x(x+4)=(x+4)(x2-4x+16)+14x(x+4).
We now have a common factor x+4, so we can factorise:
(x+4)(x2-4x+16+14x)=(x+4)(x2+10x+16)=(x+4)(x+2)(x+8).
So the roots are x=-1, -2, -4, -8.