$$
\begin{aligned}
& (a p+b q+c r+d s)^{2}+(a q-b p+c s-d r)^{2}+ \\
& +(a r-b s-c p+d q)^{2}+(a s+b r-c q-d p)^{2}
\end{aligned}
$$

Express the following as a product:

$$
\begin{aligned}
& (a p+b q+c r+d s)^{2}+(a q-b p+c s-d r)^{2}+ \\
& +(a r-b s-c p+d q)^{2}+(a s+b r-c q-d p)^{2}
\end{aligned}
$$
It is easy to notice that when the four squares are brought to polynomial form and combined, the double products all cancel out, for example, the coefficient of the $2ab$ product in the following expression: $pq - qp - rs + sr = 0$.

From each square, 4 square terms arise. Since $a, b, c, d$ each appear in the 4 expressions in some order, possibly multiplied by $(-1)$, which is irrelevant due to squaring, in the total sum the coefficient of $a^{2}, b^{2}, c^{2}, d^{2}$ is $p^{2}+q^{2}+r^{2}+s^{2}$, so the given expression can be transformed into:

$$
\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\left(p^{2}+q^{2}+r^{2}+s^{2}\right)
$$

Sándor Szalay (Debrecen, Kossuth L. Secondary School, 1st year)