Question about part b: Several other quadratic forms have this property; for example, $x^{2}+dy^{2}$ (for squarefree $d$) and $x^{2}+xy+y^{2}$. With what conditions on $a, b, c$ does the quadratic form $ax^{2}+2bxy+cy^{2}$ have the same property?

b) We have $p(x,y)=(x-2y)^{2}+(x-y)^{2}$. Now note that 1-For all $a,b\in\mathbb{Z}$ there exist $x,y\in\mathbb{Z}$ such that $a=x-2y$ and $b=x-y$. 2-For all $a,b,c,d\in\mathbb{Z}$ we have $(a^{2}+b^{2})(c^{2}+d^{2})=(ac+bd)^{2}+(ad-bc)^{2}$. We're done.