First, one can consider the rational root theorem. It states that if a polynomial (an*x^n+...+a0) has a rational root (p/q), then p is a factor of a0 and q is a factor of an. Thus, given our polynomial, p(x) = -8x^4-14*x^2+4, we have an=4 and a0=8. Thus, p=+-1, 2, 4 and q=+-1,2,4,8. So our possible roots are +-1, +-1/2, +-1/4, +-2, +-4.
Then, we can try our possible roots and find p(1/2)=0 and p(-1/2)=0. Thus, 2x+1 and 2x-1 are roots.
Then, we can divide our polynomial by 2x+1. This gives -4 x^3+2 x^2-8 x+4. Now, we can divide this by 2x-1 giving -2x^2-4. Finally, we can factor out a multiple of -2 giving us x^2+2.
Putting this all together we have our final factorization: p(x)=-2*(2x+1)*(2x-1)*(x^2+2).