Question

prove (a2+b2)(c2+d2)=(ac+bd)2 + (ad-bc)2.

Answer 1

I hope you will understand it and it's a right answer of your question.

Answer 2

$(a^2+b^2)(c^2+d^2)\\= \Big(a^2-(ib)^2\Big)\Big(c^2-(id)^2\Big)\\= (a+ib)(a-ib)(c+id)(c-id)\\=(a+ib)(c-id)(a-ib)(c+id)\\=\Big(ac+i(bc-ad)+bd\Big)\Big(ac-i(bc-ad)+bd\Big)\\=\Big((ac+bd)+i(bc-ad)\Big)\Big((ac+bd)-i(bc-ad)\Big)\\=(ac+bd)^2-\Big(i(bc-ad)\Big)^2\\=(ac+bd)^2+(bc-ad)^2. \ \blacksquare$