be careful
Square roots provide plenty of opportunities to make mistakes. So do inequalities.
There may be a much easier way that I don't see, but I would square.
If 0<a<b, then a^2<b^2
Since x+2+x^2 = (x+1/2)^2 + 7/4 > 0 , you can square to get
x^2+4x < 2xsqrt(x^23x+2)
After that you have to consider cases (x<0, x>0, x=4, x>4, x<4, etc) and square again.
Happy New Year!
