<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] Use the method of foil on the right hand side and then foil the left hand side. It's easier doing right hand side to the left hand side.
$$(ac-bd)^2=(a^2c^2-2abcd+b^2d^2)$$
$$(ad+bc)^2=(a^2d^2+2abcd+b^2c^2)$$
Then simplify. 
For the right hand side:
$$(ac-bd)^2+(ad+bc)^2= (a^2c^2-2abcd-b^2c^2)+(a^2d^2+2abcd+b^2c^2)$$
$$=a^2c^2+a^2d^2+b^2d^2+b^2c^2=RHS$$
Then take the left hand side and foil.
$$(a^2+b^2)(c^2+d^2)=a^2c^2+a^2d^2+b^2c^2+b^2d^2=LHS$$
Therefore
$RHS=LHS$ Does this make sense?
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