Herer is his solution again.
The first result is moderately interesting:
(a+b)^2 + (a-b)^2 = 2 (a^2 + b^2)
But the second one is then trivial
(a+b)^2 + (a-b)^2 + a^2 + b^2 = 3 (a^2 + b^2)
Do you have some extra constraints for the second problem to make it non-trivial?
For most folks of my generation, raised on lots and lots of factoring problems, the first problem was one of a pair that jumped off the pages of algebra books so often they were almost memorized. The sum and difference of (x+y)2 and (x-y)2 produce the results 2( x2+y2) and 4xy; both of which seemed to occur in textbook problems of my youth.
Had Gas... not already told me he had always had problems with memory I would think he had grown up on the same factoring fare and had also memorized them. Perhaps he is more like Dodgeson (Carroll) who apparently wrote about re-discovering this in his diary "towards the end of his life" as he too did not recall it. As a puzzle maker, he probably thought the addition of a third multiple would disguise a common factoring and it was this second problem that appeared in his "Pillow Problems".
Since my own memory also is very fragile, I was fortunate to find the information in Edward Wakeling's "Lewis Carroll's Games and Puzzles".
Carroll also included a somewhat more difficult probability problem in his "Pillow Problems". Since I can't make the first problem more difficult for Gas.... I will give the advanced version of this one.
An urn holds B Black and W White balls. One is selected at Random and placed in a bag without your knowledge. Then a White ball is added to this bag. A random draw from this bag produces a white ball. What is the probabilty the original ball was White?