<s>[INST] Prove that for any integers $a, b, c, d$ number $(a^2 + b^2)(c^2 + d^2)$ is a sum of two squares of integer. Prove that for any integers $a, b, c, d$ number $(a^2 + b^2)(c^2 + d^2)$ is a sum of two squares of integer.
In fact I have no idea how to do this and I'll appreciate any tips or the  solution.
 [/INST] \begin{align}
(a^2+b^2)(c^2+d^2)& =a^2c^2+a^2d^2+b^2c^2+b^2d^2 \\[10pt]
& =a^2c^2+2abcd+b^2d^2+a^2d^2-2abcd+b^2c^2 \\[10pt]
&=(ac+bd)^2+(ad-bc)^2
\end{align}
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