Why? Click on this: solve dy/dx + y^2 + 2x = 0 - Wolfram|Alpha
Why? Click on this: solve dy/dx + y^2 + 2x = 0 - Wolfram|Alpha
Your method is wrong and your solution is wrong (did you even bother to check the solution by substituting it into the original DE?)
Although satisfies , it is also required to satisfy and it doesn't. Similarly, the other part of your 'solution' has to satisfy both and ....