Factoring GCF from squared quantities

Question

hope you're all well.

Quick question: Am I allowed to factor this the way I did here?

Thanks for the help, as always! :)

Answer 1

You can use the formula for the product of two sums of squares which also gives a sum of squares to verify your operation but this could be tedious.

You have $$(A^2+B^2)(C^2+D^2)=(AC+BD)^2+(AD-BC)^2=(AD+BC)^2+(AC-BD)^2$$ But you can also use the fact that if it is true your operation, then you have an identity valid for all $(b_1,b_2,c_1,c_2)$. This second method works only if your operation is not true and you have the chance of find out an absurde equality.

Put $c_1=c_2=1$ so you have the "identity" $$(b_1^2+b_2^2-2b_2)^2+(-b_1^2-b_2^2+2b_1)^2=2(b_1^2+b_2^2)(b_1^2+b_2^2-2(b_1+b_2)+2)^2$$ Take now, for example, $b_1=3$ and $b_2=4$ so you have $b_1^2+b_2^2=5^2$. It follows the absurde $$650=8450$$ This means your operation is incorrect.