If x + y = 4 and xy = 2,

We have:

$(x+y)^{2}=x^{2}+2 x y+y^{2}$

$\Rightarrow x^{2}+y^{2}=(x+y)^{2}-2 x y$

$\Rightarrow x^{2}+y^{2}=4^{2}-2 \times 2 \quad(\because x+y=4$ and $x y=2)$

$\Rightarrow x^{2}+y^{2}=16-4$

$\Rightarrow x^{2}+y^{2}=12$