A system of any number of binomial factors each the sum of two squares being supposed multiplied together , show , by any method, that their product is the sum of two squares.
Hint: $(a^2+b^2)(p^2+q^2)=(ap-bq)^2+(aq+bp)^2$. Using this identity repeatedly will do the job.
The identity can be verified most simply by expanding each side. It is sometimes called the Brahmagupta-Fibonacci Identity.
Remark: Brahmagupta actually used a more general identity, in a study of the solutions of the so-called Pell equation. He did not call it the Pell equation, since his work was done about $1000$ years before Pell lived.
If you are familiar with complex numbers, you can note that $(a+bi)(p+qi)=(ap-bq) +i(aq+bp)$. Thus the identity can be viewed as saying that the norm of a product is the product of the norms.