2.155. Transform $\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)$ so that it becomes $(a c+b d)^{2}+(a d-b c)^{2}$.
## Solution.

Expanding the brackets, we get $a^{2} c^{2}+a^{2} d^{2}+b^{2} c^{2}+b^{2} d^{2}$. Add and subtract the expression $2 a b c d$. Then

$$
\begin{aligned}
& a^{2} c^{2}+2 a b c d+b^{2} d^{2}+a^{2} d^{2}-2 a b c d+b^{2} c^{2}=(a c+b d)^{2}+(a d-b c)^{2} \Rightarrow \\
& \Rightarrow\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=(a c+b d)^{2}+(a d-b c)^{2}
\end{aligned}
$$