I assume the two zeroes are 2+√3 and 2-√3. We multiply the two known zeroes together: (x-2-√3)(x-2+√3)=(x-2)^2-3=x^2-4x+4-3=x^2-4x+1. Now divide this quadratic into the given quartic: x^2-2x-35=(x-7)(x+5). So the other two zeroes are 7 and -5.
ALGEBRAIC LONG DIVISION
x^2-2x-35
x^2-4x+1 ) x^4-6x^3-26x^2+138x-35
- x^4-4x^3+ x^2
-2x^3-27x^2+138x
- -2x^3+ 8x^2- 2x
-35x^2+140x-35
- -35x^2+140x-35
0