4 different phenotypes may be produced. If you write this down in a Punnet square you can get the exact ratio and the genotypes along with the phenotypes.
Let us assign B for brown eyes, b for blue eyes; H for right-handed, h for left-handed.
The mother's genotype will then be bbhh
The father's genotype will then be BbHh
bh bh bh bh
BH BbHh BbHh BbHh BbHh
Bh Bbhh Bbhh Bbhh Bbhh
bH bbHh bbHh bbHh bbHh
bh bbhh bbhh bbhh bbhh
Now based on that you can see that there are 4 possible genotypic combinations which would express different phenotypic combinations as well.
BbHh: Brown-eyed, right-handed
Bbhh: Brown-eyed, left-handed
bbHh: Blue-eyed, right handed
bbhh: Blue-eyed, left-handed
So again, there would be 4 possible phenotypes.