Let p(x)=2x4-5x3-12x2+35x-14.
First look for rational zeroes which are found by dividing factors of 14 by factors of 2:
1, 2, 7, 14, ½, 7/2 (both positive and negative forms).
We need to find p(x)=0 for ± each of the above.
Start with ½: p(½)=2/16-5/8-3+35/2-14=0, so ½ is a zero.
p(1)=6; p(-1)=-54; p(2)=32-40-48+70-14=0, so 2 is a zero.
We only need two zeroes to convert the quartic into a quadratic.
Divide by the zeroes. Start with 2:
2 | 2 -5 -12 35 -14
2 4 -2 -28 | 14
2 -1 -14 7 | 0 =2x3-x2-14x+7.
Now divide by ½:
½ | 2 -1 -14 7
2 1 0 | -7
2 0 -14 | 0 =2x2-14⇒x=±√7.
So we've found all 4 zeroes: -√7, ½, 2, √7.