+2862 
Organize as follows
\((x^2+y^2)+(xy+x)+(y+1)\)
Factor as follows
\((x^2+y^2)+x(y+1)+1(y+1)\)
Re-organize as follows
\((x^2+y^2)+(x+1)(y+1)\)
We see that negative values for x and y won't make it any more minimum
We instantly know that substituting 0 will get the smallest value.
(0+0) + (1)(1)
1
The minimum value is \(\boxed{1}\)
Thanks guest for confirming answer, I just did it the complicated way so we can be sure that negative values don't decrease the minimum value
+2440 AoPS protects their images with an obfuscation code that’s active when viewed directly.
This obfuscation is easy to remedy:
For Chrome broswers: Right click on the page and select inspect. Note the three lines of code.
One of the lines will look like this: ody style="margin: 0px; background: #0e0e0e;" >
Right click on the element and select edit. Edit the hex code to “ffeeee” (no quotes), then press enter. Close the inspection dialogue box.
The page will now be easy to view.
GA
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+118608 Thanks Ginger, but AoPS protect their images for a reason.
I know a lot of kids cheat on here, or go against the instructions of their educators by gaining outside help. (not always the same thing in my view)
But I am not going to purposefully circumvent the protection that educators are trying to put in place. :)
As always though, I am impressed by your knowledge and am grateful for your attempt to help.
+2440 Yes, I see your point. The AoPS cheating issue is a problem, and has been a problem for at least three years. I’ve noticed Wonderman’s posts tagging AoPS’ homework questions, and I’ve read the hot debates among members and guests precipitated by this, along with the comments of one or more trolls directed to particular offenders. Some of these were quite funny!
This is very funny!
https://web2.0calc.com/questions/the-effects-of-cheating#r4
I can hear the laughter ....
I may address this issue further in a main thread post.
GA
