Question

If A is the set of all numbers a^2+4ab+b^2, prove that A is closed under multiplication.

I've tried a few different things, but the algebra always gets out of hand. Any help would be appreciated!

Answer 1

What have you tried?

Answer 2

I tried creating x = a^2+4ab+b^2 and y=c^2+4cd+d^2 and finding the product xy, but this gives me a mess of algebra and no clear way to find my square terms or my middle term.

I tried the same thing by rearranging:
\[a^2+4ab+b^2=(a+b)^2+2ab\](Didn't work)

And
\[a^2+4ab+b^2=2(a+b)^2-(a^2+b^2)\]Still no luck.

Thought about setting up a proof by induction since a and b are limited to the integers, but I couldn't see a clear way to do that either.