smc

First note that $x^{2}y+x{y}^2+x+y=(xy+1)(x+y)$. Substituting $6$ for $xy$ gives $7(x+y)=63$, giving a result of $x+y=9$. Squaring this equation and subtracting by $12$, gives us $x^2+y^2=\boxed{69}$