First find rational zeroes by examining the factors of the highest power term and the constant.
Since both of these are 2 which has factors 1 and 2 then the rational zeroes are:
1, 2, ½ (positive and negative versions of each).
Test each one in turn and we find that x=-½ makes f(x) zero so x=½ is a zero (root) and by dividing by it we will get a quadratic which can be solved. Use synthetic division:
-½ | 2 3 5 2
2 -1 -1 | -2
2 2 4 | 0 = 2(x2+x+2).
Solve the quadratic using the formula x=(-1±√(1-8))/2=(-1±i√7)/2.
The real factors are 2(x+½)(x2+x+2)=(2x+1)(x2+x+2).
There are two complex roots: (-1+i√7)/2 and -(1+i√7)/2.