x^{4}-2x^{3}-25x^{2}-34x-12=0.

Try rational zeroes first. So we look at the factors of 12: 1, 2, 3, 4, 6.

Start with x=1 or -1:

Clearly x=1 is going to give us a negative result, so let's try x=-1:

1+2-25+34-12=37-37=0, so x=-1, that is, x+1=0, x+1 is a factor. Divide by this factor using synthetic division by the root:

-1 | 1 -2 -25 -34 -12

__1 -1 3 22 | 12__

__1 -3 -22 -12 | 0 = x__^{3}__-3x__^{2}__-22x-12__.

Now try the next rational zero, x=3 or x=-3. x=3 clearly gives us a negative number so try x=-3:

-27-27+66-12=-66+66=0, so x=-3 is a root. Divide by the root:

-3 | 1 -3 -22 -12

__1 -3 18 | 12__

__1 -6 -4 | 0 = x__^{2}__-6x-4__

The quadratic remains to be solved.

x^{2}-6x-4=x^{2}-6x+9-13=0,

(x-3)^{2}-13=0,

x-3=±√13, x=3+√13=6.6056 or x=3-√13=-0.6056 approx.