A blue-eyed, left-handed woman marries a brown-eyed, right-handed man who is heterozygous for both of his traits. if blue eyes and left-handedness are recessive, how many different phenotypes are possible in their children?

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.