<s>[INST] How do I transform the left side into the right side of this equation? How does one transform the left side into the right side?
$$
(a^2+b^2)(c^2+d^2) = (ac-bd)^2 + (ad+bc)^2
$$
 [/INST] $$(a^2+b^2)(c^2+d^2)$$
$$=a^2.c^2+a^2.d^2+b^2.c^2+b^2.d^2$$
$$=a^2.c^2+b^2.d^2+a^2.d^2+b^2.c^2$$
$$=a^2.c^2-2a.b.c.d+b^2.d^2+a^2.d^2+2a.b.c.d+b^2.c^2$$
$$=(ac)^2 - 2.(ac).(bd)+(bd)^2 + (ad)^2 + 2(ad)(bc) + (bc)^2$$
$$=(ac - bd)^2+(ad+bc)^2$$
$$=R.H.S$$
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