Write $x(a^2+b^2)+(2ab)y$ as a product of factors. Let $a,b,c,x,y \in\mathbb{Z}>1$ and $\gcd(a,b)=\gcd(x,y)=\gcd(a,b,x,y)=1$, 
Can $$x(a^2+b^2)+(2ab)y$$be factorized ?

Since $x$ and $y$ are coprime, you can reduce the problem into proving that the squared sum of coprime integers $a$ and $b$ is coprime to $a$ and $b$, that is
$$\gcd(a^2 + b^2, a) = \gcd(a^2 + b^2, b) = 1.$$
Which can be done by considering evenness and oddness of $a$ and $b$.
But when both $a$ and $b$ are odd, $\gcd(a^2 + b^2, 2ab) = 2$. Therefore, the expression cannot be written generally as a product of factors because $a$ and $b$ aren't always both odd.
