Problem 6. Factorize

$M(a, b, c, d)=\left((a-c)^{2}+(b-d)^{2}\right)\left(a^{2}+b^{2}\right)-(a d-b c)^{2}$.
Solution. Let $a=b=1$. Then $M(1,1, c, d)=$ $=c^{2}+d^{2}+4+2 c d-4 c-4 d=(c+d-2)^{2}$.

Let $a=1, b=2$. Then $M(1,2, c, d)=c^{2}+4 d^{2}+$ $+4 c d-10 c-20 d+25=(c+2 d-5)^{2}$. But $5=2^{2}+1^{2}$. Therefore,

$$
M(1,2, c, d)=\left(c+2 d-1^{2}-2^{2}\right)^{2}
$$

The considered particular cases allow us to make the assumption that $M(a, b, c, d)=\left(a c+b d-a^{2}-b^{2}\right)^{2}$. It remains to prove this identity.